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BCH code

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In <a href="Coding_theory" title="Coding theory">coding theory</a>, the Bose–Chaudhuri–Hocquenghem codes (BCH codes) form a class of <a href="Cyclic_code" title="Cyclic code">cyclic</a> <a href="Error_correction_code" title="Error correction code">error-correcting codes</a> that are constructed using <a href="Polynomial" title="Polynomial">polynomials</a> over a <a href="Finite_field" title="Finite field">finite field</a> (also called a <a href="Finite_field" title="Finite field">Galois field</a>). BCH codes were invented in 1959 by French mathematician <a href="Alexis_Hocquenghem" title="Alexis Hocquenghem">Alexis Hocquenghem</a>, and independently in 1960 by <a href="Raj_Chandra_Bose" title="Raj Chandra Bose">Raj Chandra Bose</a> and <a href="D._K._Ray-Chaudhuri" class="mw-redirect" title="D. K. Ray-Chaudhuri">D. K. Ray-Chaudhuri</a>.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a> The name Bose–Chaudhuri–Hocquenghem (and the acronym BCH) arises from the initials of the inventors' surnames (mistakenly, in the case of Ray-Chaudhuri).

One of the key features of BCH codes is that during code design, there is a precise control over the number of symbol errors correctable by the code. In particular, it is possible to design binary BCH codes that can correct multiple bit errors. Another advantage of BCH codes is the ease with which they can be decoded, namely, via an <a href="Abstract_algebra" title="Abstract algebra">algebraic</a> method known as <a href="Syndrome_decoding" class="mw-redirect" title="Syndrome decoding">syndrome decoding</a>. This simplifies the design of the decoder for these codes, using small low-power electronic hardware.

BCH codes are used in applications such as satellite communications,<a href="#cite_note-4">[4]</a> <a href="Compact_disc" title="Compact disc">compact disc</a> players, <a href="DVD" title="DVD">DVDs</a>, <a href="Disk_storage" title="Disk storage">disk drives</a>, <a href="USB_flash_drive" title="USB flash drive">USB flash drives</a>, <a href="Solid-state_drive" title="Solid-state drive">solid-state drives</a>,<a href="#cite_note-5">[5]</a> and <a href="Barcode" title="Barcode">two-dimensional bar codes</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition_and_illustration">Definition and illustration</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Primitive_narrow-sense_BCH_codes">Primitive narrow-sense BCH codes</h3></div>

Given a <a href="Prime_number" title="Prime number">prime number</a> q and <a href="Prime_power" title="Prime power">prime power</a> qm with positive integers m and d such that d ≤ qm − 1, a primitive narrow-sense BCH code over the <a href="Finite_field" title="Finite field">finite field</a> (or Galois field) GF(q) with code length n = qm − 1 and <a href="Block_code#The_distance_d" title="Block code">distance</a> at least d is constructed by the following method.

Let α be a <a href="Simple_extension#Definition" title="Simple extension">primitive element</a> of GF(qm).

For any positive integer i, let mi(x) be the <a href="Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal polynomial</a> with coefficients in GF(q) of αi.

The <a href="Generator_polynomial" class="mw-redirect" title="Generator polynomial">generator polynomial</a> of the BCH code is defined as the <a href="Least_common_multiple" title="Least common multiple">least common multiple</a> g(x) = lcm(m1(x),…,md − 1(x)).

It can be seen that g(x) is a polynomial with coefficients in GF(q) and divides xn − 1.

Therefore, the <a href="Polynomial_code" title="Polynomial code">polynomial code</a> defined by g(x) is a cyclic code.

<div class="mw-heading mw-heading4"><h4 id="Example">Example</h4></div>

Let q = 2 and m = 4 (therefore n = 15). We will consider different values of d for GF(16) = GF(24) based on the reducing polynomial z4 + z + 1, using primitive element α(z) = z. There are fourteen minimum polynomials mi(x) with coefficients in GF(2) satisfying

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<annotation encoding="application/x-tex">{\displaystyle m_{i}\left(\alpha ^{i}\right){\bmod {\left(z^{4}+z+1\right)}}=0.}</annotation>

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</math><img src="./90ac0784b368cfd5f3f34d05068e6bcb3ec8de68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.826ex; height:3.343ex;" alt="{\displaystyle m_{i}\left(\alpha ^{i}\right){\bmod {\left(z^{4}+z+1\right)}}=0.}" loading="lazy"></dd></dl>

The minimal polynomials are

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}m_{1}(x)&=m_{2}(x)=m_{4}(x)=m_{8}(x)=x^{4}+x+1,\\m_{3}(x)&=m_{6}(x)=m_{9}(x)=m_{12}(x)=x^{4}+x^{3}+x^{2}+x+1,\\m_{5}(x)&=m_{10}(x)=x^{2}+x+1,\\m_{7}(x)&=m_{11}(x)=m_{13}(x)=m_{14}(x)=x^{4}+x^{3}+1.\end{aligned}}}</annotation>

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</math><img src="./0644e222f9fcc17cc759028363a94a57161ca3fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:60.554ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}m_{1}(x)&=m_{2}(x)=m_{4}(x)=m_{8}(x)=x^{4}+x+1,\\m_{3}(x)&=m_{6}(x)=m_{9}(x)=m_{12}(x)=x^{4}+x^{3}+x^{2}+x+1,\\m_{5}(x)&=m_{10}(x)=x^{2}+x+1,\\m_{7}(x)&=m_{11}(x)=m_{13}(x)=m_{14}(x)=x^{4}+x^{3}+1.\end{aligned}}}" loading="lazy"></dd></dl>

The BCH code with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=2,3}">

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<annotation encoding="application/x-tex">{\displaystyle d=2,3}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle g(x)={\rm {lcm}}(m_{1}(x),m_{2}(x))=m_{1}(x)=x^{4}+x+1.\,}</annotation>

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</math><img src="./554dd381d4fd3b82aad617f367a4d717abe02faa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.3ex; height:3.176ex;" alt="{\displaystyle g(x)={\rm {lcm}}(m_{1}(x),m_{2}(x))=m_{1}(x)=x^{4}+x+1.\,}" loading="lazy"></dd></dl>

It has minimal <a href="Hamming_distance" title="Hamming distance">Hamming distance</a> at least 3 and corrects up to one error. Since the generator polynomial is of degree 4, this code has 11 data bits and 4 checksum bits. It is also denoted as: (15, 11) BCH code.

The BCH code with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=4,5}">

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<annotation encoding="application/x-tex">{\displaystyle d=4,5}</annotation>

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</math><img src="./f73c83a068e165d6401a6246b7b42e78c98d55ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.673ex; height:2.509ex;" alt="{\displaystyle d=4,5}" loading="lazy"> has the generator polynomial

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g(x)&={\rm {lcm}}(m_{1}(x),m_{2}(x),m_{3}(x),m_{4}(x))=m_{1}(x)m_{3}(x)\\&=\left(x^{4}+x+1\right)\left(x^{4}+x^{3}+x^{2}+x+1\right)=x^{8}+x^{7}+x^{6}+x^{4}+1.\end{aligned}}}</annotation>

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</math><img src="./f21c678b3fd4beb09bdb6cf4d2b761349de3607b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.432ex; margin-bottom: -0.24ex; width:70.119ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}g(x)&={\rm {lcm}}(m_{1}(x),m_{2}(x),m_{3}(x),m_{4}(x))=m_{1}(x)m_{3}(x)\\&=\left(x^{4}+x+1\right)\left(x^{4}+x^{3}+x^{2}+x+1\right)=x^{8}+x^{7}+x^{6}+x^{4}+1.\end{aligned}}}" loading="lazy"></dd></dl>

It has minimal Hamming distance at least 5 and corrects up to two errors. Since the generator polynomial is of degree 8, this code has 7 data bits and 8 checksum bits. It is also denoted as: (15, 7) BCH code.

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<annotation encoding="application/x-tex">{\displaystyle d=6,7}</annotation>

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</math><img src="./876bd68bab0688a2c9cbdf661a73f7ff3c2f62e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.673ex; height:2.509ex;" alt="{\displaystyle d=6,7}" loading="lazy"> has the generator polynomial

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g(x)&={\rm {lcm}}(m_{1}(x),m_{2}(x),m_{3}(x),m_{4}(x),m_{5}(x),m_{6}(x))=m_{1}(x)m_{3}(x)m_{5}(x)\\&=\left(x^{4}+x+1\right)\left(x^{4}+x^{3}+x^{2}+x+1\right)\left(x^{2}+x+1\right)=x^{10}+x^{8}+x^{5}+x^{4}+x^{2}+x+1.\end{aligned}}}</annotation>

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</math><img src="./1c8f24a57e47baa1739c3f6a6fdf2e46146f185e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.432ex; margin-bottom: -0.24ex; width:93.409ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}g(x)&={\rm {lcm}}(m_{1}(x),m_{2}(x),m_{3}(x),m_{4}(x),m_{5}(x),m_{6}(x))=m_{1}(x)m_{3}(x)m_{5}(x)\\&=\left(x^{4}+x+1\right)\left(x^{4}+x^{3}+x^{2}+x+1\right)\left(x^{2}+x+1\right)=x^{10}+x^{8}+x^{5}+x^{4}+x^{2}+x+1.\end{aligned}}}" loading="lazy"></dd></dl>

It has minimal Hamming distance at least 7 and corrects up to three errors. Since the generator polynomial is of degree 10, this code has 5 data bits and 10 checksum bits. It is also denoted as: (15, 5) BCH code. (This particular generator polynomial has a real-world application, in the "format information" of the <a href="QR_code" title="QR code">QR code</a>.)

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<annotation encoding="application/x-tex">{\displaystyle d=8}</annotation>

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</math><img src="./2533106c0680f57c4c5313d8ae62484bd20aa832.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.176ex;" alt="{\displaystyle d=8}" loading="lazy"> and higher has the generator polynomial

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g(x)&={\rm {lcm}}(m_{1}(x),m_{2}(x),...,m_{14}(x))=m_{1}(x)m_{3}(x)m_{5}(x)m_{7}(x)\\&=\left(x^{4}+x+1\right)\left(x^{4}+x^{3}+x^{2}+x+1\right)\left(x^{2}+x+1\right)\left(x^{4}+x^{3}+1\right)=x^{14}+x^{13}+x^{12}+\cdots +x^{2}+x+1.\end{aligned}}}</annotation>

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</math><img src="./1f761ffc5862338ca5ee2b2de5d306e4203e04ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.432ex; margin-bottom: -0.24ex; width:109.52ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}g(x)&={\rm {lcm}}(m_{1}(x),m_{2}(x),...,m_{14}(x))=m_{1}(x)m_{3}(x)m_{5}(x)m_{7}(x)\\&=\left(x^{4}+x+1\right)\left(x^{4}+x^{3}+x^{2}+x+1\right)\left(x^{2}+x+1\right)\left(x^{4}+x^{3}+1\right)=x^{14}+x^{13}+x^{12}+\cdots +x^{2}+x+1.\end{aligned}}}" loading="lazy"></dd></dl>

This code has minimal Hamming distance 15 and corrects 7 errors. It has 1 data bit and 14 checksum bits. It is also denoted as: (15, 1) BCH code. In fact, this code has only two codewords: 000000000000000 and 111111111111111 (a trivial <a href="Repetition_code" title="Repetition code">repetition code</a>).

<div class="mw-heading mw-heading3"><h3 id="General_BCH_codes">General BCH codes</h3></div>

General BCH codes differ from primitive narrow-sense BCH codes in two respects.

First, the requirement that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (q^{m})}</annotation>

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</math><img src="./85695952b26d6e22262466201b2e54e532b8c5e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.906ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (q^{m})}" loading="lazy"> can be relaxed. By relaxing this requirement, the code length changes from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{m}-1}">

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<annotation encoding="application/x-tex">{\displaystyle q^{m}-1}</annotation>

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</math><img src="./91ab9416f8f7a6f9a329c6ccd24524f163bf628c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.757ex; height:2.676ex;" alt="{\displaystyle q^{m}-1}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ord} (\alpha ),}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {ord} (\alpha ),}</annotation>

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</math><img src="./4e1e9b6c1f4abc066a9918689762c64f9041aa4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.31ex; height:2.843ex;" alt="{\displaystyle \mathrm {ord} (\alpha ),}" loading="lazy"> the <a href="Order_(group_theory)" title="Order (group theory)">order</a> of the element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha .}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha .}</annotation>

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</math><img src="./794850adc0db51d11a6d8cfa857538183424909c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.134ex; height:1.676ex;" alt="{\displaystyle \alpha .}" loading="lazy">

Second, the consecutive roots of the generator polynomial may run from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2}}</annotation>

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</math><img src="./298fa69786fd72f6bea62e3a4c651dce1aecf38e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.281ex; height:3.009ex;" alt="{\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2}}" loading="lazy"> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\ldots ,\alpha ^{d-1}.}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha ,\ldots ,\alpha ^{d-1}.}</annotation>

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Definition. Fix a finite field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GF(q),}">

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<annotation encoding="application/x-tex">{\displaystyle GF(q),}</annotation>

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</math><img src="./a95fd0512e7960fcd72d484b6de633afbad425cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.093ex; height:2.843ex;" alt="{\displaystyle GF(q),}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> is a prime power. Choose positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n,d,c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m,n,d,c}</annotation>

</semantics>

</math><img src="./811b244a721216690c17b3fcb5605e143d8c139c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.76ex; height:2.509ex;" alt="{\displaystyle m,n,d,c}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\leq d\leq n,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\leq d\leq n,}</annotation>

</semantics>

</math><img src="./8b9562b685c447fdc2aa61f1a726ecb880be7423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.617ex; height:2.509ex;" alt="{\displaystyle 2\leq d\leq n,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {gcd}}(n,q)=1,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\rm {gcd}}(n,q)=1,}</annotation>

</semantics>

</math><img src="./fa011bf81d66800b30c8817ccf9676805fa83e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.703ex; height:2.843ex;" alt="{\displaystyle {\rm {gcd}}(n,q)=1,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> is the <a href="Multiplicative_order" title="Multiplicative order">multiplicative order</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n.}</annotation>

</semantics>

</math><img src="./e59df02a9f67a5da3c220f1244c99a46cc4eb1c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:1.676ex;" alt="{\displaystyle n.}" loading="lazy">

As before, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> be a <a href="Primitive_nth_root_of_unity" class="mw-redirect" title="Primitive nth root of unity">primitive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">th root of unity</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GF(q^{m}),}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle GF(q^{m}),}</annotation>

</semantics>

</math><img src="./d9027fe0f958b3a43e1dafc983293b49a0b347ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.778ex; height:2.843ex;" alt="{\displaystyle GF(q^{m}),}" loading="lazy"> and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{i}(x)}</annotation>

</semantics>

</math><img src="./deda173f8a2ac08aafe182ce383c88f18e9ff64a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.979ex; height:2.843ex;" alt="{\displaystyle m_{i}(x)}" loading="lazy"> be the <a href="Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal polynomial</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GF(q)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle GF(q)}</annotation>

</semantics>

</math><img src="./7632e02672649089c63c998757aa3afc0782a0f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.446ex; height:2.843ex;" alt="{\displaystyle GF(q)}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{i}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{i}}</annotation>

</semantics>

</math><img src="./c114e2fb3c65eea83541fc81d81ab6df7fca4da2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.287ex; height:2.676ex;" alt="{\displaystyle \alpha ^{i}}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i.}</annotation>

</semantics>

</math><img src="./6ffcf9ad7ad44f04fa43c5b604b4801e089981cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.449ex; height:2.176ex;" alt="{\displaystyle i.}" loading="lazy">

The generator polynomial of the BCH code is defined as the <a href="Least_common_multiple" title="Least common multiple">least common multiple</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)={\rm {lcm}}(m_{c}(x),\ldots ,m_{c+d-2}(x)).}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)={\rm {lcm}}(m_{c}(x),\ldots ,m_{c+d-2}(x)).}</annotation>

</semantics>

</math><img src="./8ac516fe276b7c12f8ee1b686cbbcb3178eb01bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.089ex; height:2.843ex;" alt="{\displaystyle g(x)={\rm {lcm}}(m_{c}(x),\ldots ,m_{c+d-2}(x)).}" loading="lazy">

Note: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=q^{m}-1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=q^{m}-1}</annotation>

</semantics>

</math><img src="./6f0ba9e17d688cdf4fb87e163af43f9c590bd5fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.25ex; height:2.676ex;" alt="{\displaystyle n=q^{m}-1}" loading="lazy"> as in the simplified definition, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {gcd}}(n,q)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\rm {gcd}}(n,q)}</annotation>

</semantics>

</math><img src="./ed5ee43d2bbc76b536a4cffad17fab4142bc73ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.795ex; height:2.843ex;" alt="{\displaystyle {\rm {gcd}}(n,q)}" loading="lazy"> is 1, and the order of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m.}</annotation>

</semantics>

</math><img src="./5bd92c867d56467c0f878ef318eefcd701b8ec1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.687ex; height:1.676ex;" alt="{\displaystyle m.}" loading="lazy">

Therefore, the simplified definition is indeed a special case of the general one.

<div class="mw-heading mw-heading3"><h3 id="Special_cases">Special cases</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=1}</annotation>

</semantics>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=q^{m}-1}</annotation>

</semantics>

The generator polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation>

</semantics>

</math><img src="./c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" loading="lazy"> of a BCH code has coefficients from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GF} (q).}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (q).}</annotation>

</semantics>

</math><img src="./cae3f40124fda142ecfb02e8ee5aced684eab9da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.868ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (q).}" loading="lazy">

In general, a cyclic code over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GF} (q^{p})}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (q^{p})}</annotation>

</semantics>

</math><img src="./98c65c556e9a000acfcaec0bbcb44c6d85040db1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.29ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (q^{p})}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation>

</semantics>

</math><img src="./c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" loading="lazy"> as the generator polynomial is called a BCH code over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GF} (q^{p}).}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (q^{p}).}</annotation>

</semantics>

</math><img src="./277b2e8f98b1a0f28092d3d649c1040cd665a53a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.937ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (q^{p}).}" loading="lazy">

The BCH code over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GF} (q^{m})}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (q^{m})}</annotation>

</semantics>

</math><img src="./85695952b26d6e22262466201b2e54e532b8c5e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.906ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (q^{m})}" loading="lazy"> and generator polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation>

</semantics>

</math><img src="./c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" loading="lazy"> with successive powers of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> as roots is one type of <a href="Reed%E2%80%93Solomon_code" class="mw-redirect" title="Reed–Solomon code">Reed–Solomon code</a> where the decoder (syndromes) alphabet is the same as the channel (data and generator polynomial) alphabet, all elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GF} (q^{m})}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (q^{m})}</annotation>

</semantics>

</math><img src="./85695952b26d6e22262466201b2e54e532b8c5e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.906ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (q^{m})}" loading="lazy"> .<a href="#cite_note-6">[6]</a> The other type of Reed Solomon code is an <a href="Reed%E2%80%93Solomon_error_correction#Reed_&_Solomon's_original_view:_The_codeword_as_a_sequence_of_values" title="Reed–Solomon error correction">original view Reed Solomon code</a> which is not a BCH code.

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

The generator polynomial of a BCH code has degree at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (d-1)m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (d-1)m}</annotation>

</semantics>

</math><img src="./f1de126ccf5ea0fc49bbf2cb21528346b310c5a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.068ex; height:2.843ex;" alt="{\displaystyle (d-1)m}" loading="lazy">. Moreover, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=2}</annotation>

</semantics>

</math><img src="./26622af6012fb982cab4e9584f57dd4f364233b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=2}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=1}</annotation>

</semantics>

</math><img src="./3e3467f9e219a5ea38a30da5c3a02c2c23f61a79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c=1}" loading="lazy">, the generator polynomial has degree at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dm/2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dm/2}</annotation>

</semantics>

</math><img src="./dbc16c58aee376806de6e7efc870e3e1c23e0a2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.581ex; height:2.843ex;" alt="{\displaystyle dm/2}" loading="lazy">.

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</style>

<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Proof</div>

</th></tr>

<tr>

Each minimal polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{i}(x)}</annotation>

</semantics>

</math><img src="./deda173f8a2ac08aafe182ce383c88f18e9ff64a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.979ex; height:2.843ex;" alt="{\displaystyle m_{i}(x)}" loading="lazy"> has degree at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">. Therefore, the least common multiple of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d-1}</annotation>

</semantics>

</math><img src="./0195b64ba44bcc80b4c98e9d34256b4043fe519e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.219ex; height:2.343ex;" alt="{\displaystyle d-1}" loading="lazy"> of them has degree at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (d-1)m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (d-1)m}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=2,}</annotation>

</semantics>

</math><img src="./c75fcaa42525c0979b76e6075bac1ceedeebbbf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.977ex; height:2.509ex;" alt="{\displaystyle q=2,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}(x)=m_{2i}(x)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{i}(x)=m_{2i}(x)}</annotation>

</semantics>

</math><img src="./25f129a22b90baf6d01714deaed8f011c348b966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.879ex; height:2.843ex;" alt="{\displaystyle m_{i}(x)=m_{2i}(x)}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">. Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation>

</semantics>

</math><img src="./c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" loading="lazy"> is the least common multiple of at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d/2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d/2}</annotation>

</semantics>

</math><img src="./582b6455b1ff5f4fb027024a8b1458687dc8ed74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.541ex; height:2.843ex;" alt="{\displaystyle d/2}" loading="lazy"> minimal polynomials <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{i}(x)}</annotation>

</semantics>

</math><img src="./deda173f8a2ac08aafe182ce383c88f18e9ff64a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.979ex; height:2.843ex;" alt="{\displaystyle m_{i}(x)}" loading="lazy"> for odd indices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i,}</annotation>

</semantics>

</math><img src="./d3d0f7dadba3056fa3c06a6bee5c0b4182471152.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle i,}" loading="lazy"> each of degree at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</td></tr></tbody></table></div>

A BCH code has minimal Hamming distance at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy">.

<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Proof</div>

</th></tr>

<tr>

Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)}</annotation>

</semantics>

</math><img src="./8cb7afced134ef75572e5314a5d278c2d644f438.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}" loading="lazy"> is a code word with fewer than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<mtext> where </mtext>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)=b_{1}x^{k_{1}}+\cdots +b_{d-1}x^{k_{d-1}},{\text{ where }}k_{1}<k_{2}<\cdots <k_{d-1}.}</annotation>

</semantics>

</math><img src="./4b6ac3d7dbf98f66a6ddd80373299eb4e0783988.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:60.121ex; height:3.176ex;" alt="{\displaystyle p(x)=b_{1}x^{k_{1}}+\cdots +b_{d-1}x^{k_{d-1}},{\text{ where }}k_{1}<k_{2}<\cdots <k_{d-1}.}" loading="lazy"></dd></dl>

Recall that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2}}</annotation>

</semantics>

</math><img src="./298fa69786fd72f6bea62e3a4c651dce1aecf38e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.281ex; height:3.009ex;" alt="{\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2}}" loading="lazy"> are roots of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x),}</annotation>

</semantics>

</math><img src="./8c36278728763484856f51a229bf8ce5e592e61e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.902ex; height:2.843ex;" alt="{\displaystyle g(x),}" loading="lazy"> hence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)}</annotation>

</semantics>

</math><img src="./8cb7afced134ef75572e5314a5d278c2d644f438.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}" loading="lazy">. This implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1},\ldots ,b_{d-1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{1},\ldots ,b_{d-1}}</annotation>

</semantics>

</math><img src="./467edfcfd87ba78557f633ac9ffdedf320167474.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.42ex; height:2.509ex;" alt="{\displaystyle b_{1},\ldots ,b_{d-1}}" loading="lazy"> satisfy the following equations, for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \{c,\dotsc ,c+d-2\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i\in \{c,\dotsc ,c+d-2\}}</annotation>

</semantics>

</math><img src="./33535289ebf09ca3f152b40adfc9b0bbf3bdd4f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.219ex; height:2.843ex;" alt="{\displaystyle i\in \{c,\dotsc ,c+d-2\}}" loading="lazy">:

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(\alpha ^{i})=b_{1}\alpha ^{ik_{1}}+b_{2}\alpha ^{ik_{2}}+\cdots +b_{d-1}\alpha ^{ik_{d-1}}=0.}</annotation>

</semantics>

</math><img src="./e9bfd234a551f9c5ea788a29837c3583a9aecbbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:46.562ex; height:3.176ex;" alt="{\displaystyle p(\alpha ^{i})=b_{1}\alpha ^{ik_{1}}+b_{2}\alpha ^{ik_{2}}+\cdots +b_{d-1}\alpha ^{ik_{d-1}}=0.}" loading="lazy"></dd></dl>

In matrix form, we have

<mrow>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\alpha ^{ck_{1}}&\alpha ^{ck_{2}}&\cdots &\alpha ^{ck_{d-1}}\\\alpha ^{(c+1)k_{1}}&\alpha ^{(c+1)k_{2}}&\cdots &\alpha ^{(c+1)k_{d-1}}\\\vdots &\vdots &&\vdots \\\alpha ^{(c+d-2)k_{1}}&\alpha ^{(c+d-2)k_{2}}&\cdots &\alpha ^{(c+d-2)k_{d-1}}\\\end{bmatrix}}{\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{d-1}\end{bmatrix}}={\begin{bmatrix}0\\0\\\vdots \\0\end{bmatrix}}.}</annotation>

</semantics>

</math><img src="./7c7ccb2c99bd7b3129dd6ca95f840446dac47146.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:60.995ex; height:14.176ex;" alt="{\displaystyle {\begin{bmatrix}\alpha ^{ck_{1}}&\alpha ^{ck_{2}}&\cdots &\alpha ^{ck_{d-1}}\\\alpha ^{(c+1)k_{1}}&\alpha ^{(c+1)k_{2}}&\cdots &\alpha ^{(c+1)k_{d-1}}\\\vdots &\vdots &&\vdots \\\alpha ^{(c+d-2)k_{1}}&\alpha ^{(c+d-2)k_{2}}&\cdots &\alpha ^{(c+d-2)k_{d-1}}\\\end{bmatrix}}{\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{d-1}\end{bmatrix}}={\begin{bmatrix}0\\0\\\vdots \\0\end{bmatrix}}.}" loading="lazy"></dd></dl>

The determinant of this matrix equals

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(\prod _{i=1}^{d-1}\alpha ^{ck_{i}}\right)\det {\begin{pmatrix}1&1&\cdots &1\\\alpha ^{k_{1}}&\alpha ^{k_{2}}&\cdots &\alpha ^{k_{d-1}}\\\vdots &\vdots &&\vdots \\\alpha ^{(d-2)k_{1}}&\alpha ^{(d-2)k_{2}}&\cdots &\alpha ^{(d-2)k_{d-1}}\\\end{pmatrix}}=\left(\prod _{i=1}^{d-1}\alpha ^{ck_{i}}\right)\det(V).}</annotation>

</semantics>

</math><img src="./20cde491b5e4adbb6c793aae6acd01aa2e82230a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:76.052ex; height:14.176ex;" alt="{\displaystyle \left(\prod _{i=1}^{d-1}\alpha ^{ck_{i}}\right)\det {\begin{pmatrix}1&1&\cdots &1\\\alpha ^{k_{1}}&\alpha ^{k_{2}}&\cdots &\alpha ^{k_{d-1}}\\\vdots &\vdots &&\vdots \\\alpha ^{(d-2)k_{1}}&\alpha ^{(d-2)k_{2}}&\cdots &\alpha ^{(d-2)k_{d-1}}\\\end{pmatrix}}=\left(\prod _{i=1}^{d-1}\alpha ^{ck_{i}}\right)\det(V).}" loading="lazy"></dd></dl>

The matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> is seen to be a <a href="Vandermonde_matrix" title="Vandermonde matrix">Vandermonde matrix</a>, and its determinant is

</mrow>

</munder>

<mrow>

<mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det(V)=\prod _{1\leq i<j\leq d-1}\left(\alpha ^{k_{j}}-\alpha ^{k_{i}}\right),}</annotation>

</semantics>

</math><img src="./f21ed729de64bbc1d4e185b884e42af513c441e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:31.67ex; height:5.843ex;" alt="{\displaystyle \det(V)=\prod _{1\leq i<j\leq d-1}\left(\alpha ^{k_{j}}-\alpha ^{k_{i}}\right),}" loading="lazy"></dd></dl>

which is non-zero. It therefore follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1},\ldots ,b_{d-1}=0,}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{1},\ldots ,b_{d-1}=0,}</annotation>

</semantics>

</math><img src="./1de5ca3b22f3083070d3bf36a0d005ca98c290c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.328ex; height:2.509ex;" alt="{\displaystyle b_{1},\ldots ,b_{d-1}=0,}" loading="lazy"> hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)=0.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)=0.}</annotation>

</semantics>

</math><img src="./4a571802d84fd7c04368fa12df95c18259feb1da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.306ex; height:2.843ex;" alt="{\displaystyle p(x)=0.}" loading="lazy">

</td></tr></tbody></table></div>

A BCH code is cyclic.

<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Proof</div>

</th></tr>

<tr>

A polynomial code of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is cyclic if and only if its generator polynomial divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}-1.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{n}-1.}</annotation>

</semantics>

</math><img src="./a8d5639cd7157fe06c622e0509fdec62d5e65e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle x^{n}-1.}" loading="lazy"> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation>

</semantics>

</math><img src="./c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" loading="lazy"> is the minimal polynomial with roots <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2},}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2},}</annotation>

</semantics>

</math><img src="./8f714a3261d08707485a4c386de13d3d449a46ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.928ex; height:3.009ex;" alt="{\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2},}" loading="lazy"> it suffices to check that each of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2}}</annotation>

</semantics>

</math><img src="./298fa69786fd72f6bea62e3a4c651dce1aecf38e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.281ex; height:3.009ex;" alt="{\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2}}" loading="lazy"> is a root of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}-1.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{n}-1.}</annotation>

</semantics>

</math><img src="./a8d5639cd7157fe06c622e0509fdec62d5e65e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle x^{n}-1.}" loading="lazy"> This follows immediately from the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</td></tr></tbody></table></div>

<div class="mw-heading mw-heading2"><h2 id="Encoding">Encoding</h2></div>

Because any polynomial that is a multiple of the generator polynomial is a valid BCH codeword, BCH encoding is merely the process of finding some polynomial that has the generator as a factor.

The BCH code itself is not prescriptive about the meaning of the coefficients of the polynomial; conceptually, a BCH decoding algorithm's sole concern is to find the valid codeword with the minimal Hamming distance to the received codeword. Therefore, the BCH code may be implemented either as a <a href="Systematic_code" title="Systematic code">systematic code</a> or not, depending on how the implementor chooses to embed the message in the encoded polynomial.

<div class="mw-heading mw-heading3"><h3 id="Non-systematic_encoding:_The_message_as_a_factor">Non-systematic encoding: The message as a factor</h3></div>

The most straightforward way to find a polynomial that is a multiple of the generator is to compute the product of some arbitrary polynomial and the generator. In this case, the arbitrary polynomial can be chosen using the symbols of the message as coefficients.

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(x)=p(x)g(x)}</annotation>

</semantics>

</math><img src="./3158d120df065bf2602fba044f9a93c886f6107c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.891ex; height:2.843ex;" alt="{\displaystyle s(x)=p(x)g(x)}" loading="lazy"></dd></dl>

As an example, consider the generator polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=x^{10}+x^{9}+x^{8}+x^{6}+x^{5}+x^{3}+1}">

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<annotation encoding="application/x-tex">{\displaystyle g(x)=x^{10}+x^{9}+x^{8}+x^{6}+x^{5}+x^{3}+1}</annotation>

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</math><img src="./b1926cfe961ed6090ae384ab0eab2832a6d50c17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.684ex; height:3.176ex;" alt="{\displaystyle g(x)=x^{10}+x^{9}+x^{8}+x^{6}+x^{5}+x^{3}+1}" loading="lazy">, chosen for use in the (31, 21) binary BCH code used by <a href="POCSAG" class="mw-redirect" title="POCSAG">POCSAG</a> and others. To encode the 21-bit message {101101110111101111101}, we first represent it as a polynomial over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GF(2)}">

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<annotation encoding="application/x-tex">{\displaystyle GF(2)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle p(x)=x^{20}+x^{18}+x^{17}+x^{15}+x^{14}+x^{13}+x^{11}+x^{10}+x^{9}+x^{8}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+1}</annotation>

</semantics>

</math><img src="./bfd5e2585ce53aa41dea00ced06587185fc73521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:93.599ex; height:3.176ex;" alt="{\displaystyle p(x)=x^{20}+x^{18}+x^{17}+x^{15}+x^{14}+x^{13}+x^{11}+x^{10}+x^{9}+x^{8}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+1}" loading="lazy"></dd></dl>

Then, compute (also over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GF(2)}">

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<annotation encoding="application/x-tex">{\displaystyle GF(2)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s(x)&=p(x)g(x)\\&=\left(x^{20}+x^{18}+x^{17}+x^{15}+x^{14}+x^{13}+x^{11}+x^{10}+x^{9}+x^{8}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+1\right)\left(x^{10}+x^{9}+x^{8}+x^{6}+x^{5}+x^{3}+1\right)\\&=x^{30}+x^{29}+x^{26}+x^{25}+x^{24}+x^{22}+x^{19}+x^{17}+x^{16}+x^{15}+x^{14}+x^{12}+x^{10}+x^{9}+x^{8}+x^{6}+x^{5}+x^{4}+x^{2}+1\end{aligned}}}</annotation>

</semantics>

</math><img src="./998a57dbc543278964375b4c04821af07e655088.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:132.16ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}s(x)&=p(x)g(x)\\&=\left(x^{20}+x^{18}+x^{17}+x^{15}+x^{14}+x^{13}+x^{11}+x^{10}+x^{9}+x^{8}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+1\right)\left(x^{10}+x^{9}+x^{8}+x^{6}+x^{5}+x^{3}+1\right)\\&=x^{30}+x^{29}+x^{26}+x^{25}+x^{24}+x^{22}+x^{19}+x^{17}+x^{16}+x^{15}+x^{14}+x^{12}+x^{10}+x^{9}+x^{8}+x^{6}+x^{5}+x^{4}+x^{2}+1\end{aligned}}}" loading="lazy"></dd></dl>

Thus, the transmitted codeword is {1100111010010111101011101110101}.

The receiver can use these bits as coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(x)}</annotation>

</semantics>

</math><img src="./10f51eec88706f1a26ac7430dc71d92c15e71a6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.229ex; height:2.843ex;" alt="{\displaystyle s(x)}" loading="lazy"> and, after error-correction to ensure a valid codeword, can recompute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)=s(x)/g(x)}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle p(x)=s(x)/g(x)}</annotation>

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</math><img src="./ec30aa50fab20ab5b70348d1fbd6dc13bf544c5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:17.143ex; height:2.843ex;" alt="{\displaystyle p(x)=s(x)/g(x)}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Systematic_encoding:_The_message_as_a_prefix">Systematic encoding: The message as a prefix</h3></div>

A systematic code is one in which the message appears verbatim somewhere within the codeword. Therefore, systematic BCH encoding involves first embedding the message polynomial within the codeword polynomial, and then adjusting the coefficients of the remaining (non-message) terms to ensure that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle s(x)}</annotation>

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</math><img src="./10f51eec88706f1a26ac7430dc71d92c15e71a6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.229ex; height:2.843ex;" alt="{\displaystyle s(x)}" loading="lazy"> is divisible by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation>

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This encoding method leverages the fact that subtracting the remainder from a dividend results in a multiple of the divisor. Hence, if we take our message polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x)}</annotation>

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</math><img src="./8cb7afced134ef75572e5314a5d278c2d644f438.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}" loading="lazy"> as before and multiply it by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n-k}}">

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<annotation encoding="application/x-tex">{\displaystyle x^{n-k}}</annotation>

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</math><img src="./a8070b46f1e0f725066382bdec6813c614218db7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.683ex; height:2.676ex;" alt="{\displaystyle x^{n-k}}" loading="lazy"> (to "shift" the message out of the way of the remainder), we can then use <a href="Euclidean_division" title="Euclidean division">Euclidean division</a> of polynomials to yield:

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<annotation encoding="application/x-tex">{\displaystyle p(x)x^{n-k}=q(x)g(x)+r(x)}</annotation>

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</math><img src="./58be78cc2a36f4160fd0db28e6dc02958d0b8417.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:27.671ex; height:3.176ex;" alt="{\displaystyle p(x)x^{n-k}=q(x)g(x)+r(x)}" loading="lazy"></dd></dl>

Here, we see that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x)g(x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle q(x)g(x)}</annotation>

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</math><img src="./e4209847b394252bdebf5e9b4df189bb36a05b71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.464ex; height:2.843ex;" alt="{\displaystyle q(x)g(x)}" loading="lazy"> is a valid codeword. As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(x)}">

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<annotation encoding="application/x-tex">{\displaystyle r(x)}</annotation>

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</math><img src="./0d457e88abd861f46ede37d27c4669c747367109.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.188ex; height:2.843ex;" alt="{\displaystyle r(x)}" loading="lazy"> is always of degree less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-k}">

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<annotation encoding="application/x-tex">{\displaystyle n-k}</annotation>

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</math><img src="./b98e1d6a69bccd09a4b9b69bdf03a08c1706c8c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.446ex; height:2.343ex;" alt="{\displaystyle n-k}" loading="lazy"> (which is the degree of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">

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<annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation>

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</math><img src="./c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" loading="lazy">), we can safely subtract it from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)x^{n-k}}">

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<annotation encoding="application/x-tex">{\displaystyle p(x)x^{n-k}}</annotation>

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</math><img src="./4e7b757ad9ea8f7f16c7b61fa4ed73d47ae25dc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.081ex; height:3.176ex;" alt="{\displaystyle p(x)x^{n-k}}" loading="lazy"> without altering any of the message coefficients, hence we have our <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)}">

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<annotation encoding="application/x-tex">{\displaystyle s(x)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle s(x)=q(x)g(x)=p(x)x^{n-k}-r(x)}</annotation>

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</math><img src="./1fb14d707e7484ce00ddcfabedac5d73ce46b58e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.909ex; height:3.176ex;" alt="{\displaystyle s(x)=q(x)g(x)=p(x)x^{n-k}-r(x)}" loading="lazy"></dd></dl>

Over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GF(2)}">

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<annotation encoding="application/x-tex">{\displaystyle GF(2)}</annotation>

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</math><img src="./7b0f9e0401a155f4a1fe47de571a153078552e44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.539ex; height:2.843ex;" alt="{\displaystyle GF(2)}" loading="lazy"> (i.e. with binary BCH codes), this process is indistinguishable from appending a <a href="Cyclic_redundancy_check" title="Cyclic redundancy check">cyclic redundancy check</a>, and if a systematic binary BCH code is used only for error-detection purposes, we see that BCH codes are just a generalization of the <a href="Mathematics_of_cyclic_redundancy_checks" title="Mathematics of cyclic redundancy checks">mathematics of cyclic redundancy checks</a>.

The advantage to the systematic coding is that the receiver can recover the original message by discarding everything after the first <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> coefficients, after performing error correction.

<div class="mw-heading mw-heading2"><h2 id="Decoding">Decoding</h2></div>

There are many algorithms for decoding BCH codes. The most common ones follow this general outline:

<ol><li>Calculate the syndromes sj for the received vector</li>

<li>Determine the number of errors t and the error locator polynomial Λ(x) from the syndromes</li>

<li>Calculate the roots of the error location polynomial to find the error locations Xi</li>

<li>Calculate the error values Yi at those error locations</li>

<li>Correct the errors</li></ol>

During some of these steps, the decoding algorithm may determine that the received vector has too many errors and cannot be corrected. For example, if an appropriate value of t is not found, then the correction would fail. In a truncated (not primitive) code, an error location may be out of range. If the received vector has more errors than the code can correct, the decoder may unknowingly produce an apparently valid message that is not the one that was sent.

<div class="mw-heading mw-heading3"><h3 id="Calculate_the_syndromes">Calculate the syndromes</h3></div>

The received vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> is the sum of the correct codeword <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> and an unknown error vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E.}">

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<annotation encoding="application/x-tex">{\displaystyle E.}</annotation>

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</math><img src="./4a2566d01f104ef084ea424b8b35c2534f7f902b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.422ex; height:2.176ex;" alt="{\displaystyle E.}" loading="lazy"> The syndrome values are formed by considering <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> as a polynomial and evaluating it at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2}.}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2}.}</annotation>

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</math><img src="./26dae266f7c830257eb4b7149dbea20ae57c04ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.928ex; height:3.009ex;" alt="{\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2}.}" loading="lazy"> Thus the syndromes are<a href="#cite_note-7">[7]</a>

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<annotation encoding="application/x-tex">{\displaystyle s_{j}=R\left(\alpha ^{j}\right)=C\left(\alpha ^{j}\right)+E\left(\alpha ^{j}\right)}</annotation>

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</math><img src="./90d9ded8d20faec83c167e171c0bcaece09b31e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.086ex; height:3.343ex;" alt="{\displaystyle s_{j}=R\left(\alpha ^{j}\right)=C\left(\alpha ^{j}\right)+E\left(\alpha ^{j}\right)}" loading="lazy"></dd></dl>

for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=c}">

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</math><img src="./8fefeb9e59f2b9c8e27c975fa72a7abd833e6e6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.09ex; height:2.509ex;" alt="{\displaystyle j=c}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c+d-2.}">

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<annotation encoding="application/x-tex">{\displaystyle c+d-2.}</annotation>

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</math><img src="./3f3f90ab39ee5988e3867e4792eead5e8f29e92b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.713ex; height:2.343ex;" alt="{\displaystyle c+d-2.}" loading="lazy">

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{j}}">

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</math><img src="./27b185f9f9058e5b7f59374f9a0e2069f9d0c237.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.397ex; height:2.676ex;" alt="{\displaystyle \alpha ^{j}}" loading="lazy"> are the zeros of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x),}">

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</math><img src="./8c36278728763484856f51a229bf8ce5e592e61e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.902ex; height:2.843ex;" alt="{\displaystyle g(x),}" loading="lazy"> of which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(x)}">

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</math><img src="./7f6b53682fddbe3028ffd75bd75771b86c6c7bd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.905ex; height:2.843ex;" alt="{\displaystyle C(x)}" loading="lazy"> is a multiple, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\left(\alpha ^{j}\right)=0.}">

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<annotation encoding="application/x-tex">{\displaystyle C\left(\alpha ^{j}\right)=0.}</annotation>

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</math><img src="./44457e177097362f562662648ceb2b3fb1e41d1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.588ex; height:3.343ex;" alt="{\displaystyle C\left(\alpha ^{j}\right)=0.}" loading="lazy"> Examining the syndrome values thus isolates the error vector so one can begin to solve for it.

If there is no error, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{j}=0}">

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</math><img src="./79258d01ad636a41b377c1302966e14a256ada1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.261ex; height:2.843ex;" alt="{\displaystyle s_{j}=0}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j.}">

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</math><img src="./3a24040d2c50c228edf9b031ce3db3d04101cb22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:1.632ex; height:2.509ex;" alt="{\displaystyle j.}" loading="lazy"> If the syndromes are all zero, then the decoding is done.

<div class="mw-heading mw-heading3"><h3 id="Calculate_the_error_location_polynomial">Calculate the error location polynomial</h3></div>

If there are nonzero syndromes, then there are errors. The decoder needs to figure out how many errors and the location of those errors.

If there is a single error, write this as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(x)=e\,x^{i},}">

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</math><img src="./ed2b33ebf5d4e0196b002c73ace3d8ce56c58cdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.26ex; height:3.176ex;" alt="{\displaystyle E(x)=e\,x^{i},}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> is the location of the error and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s_{c}&=e\,\alpha ^{c\,i}\\s_{c+1}&=e\,\alpha ^{(c+1)\,i}=\alpha ^{i}s_{c}\end{aligned}}}</annotation>

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</math><img src="./dddd0d28523f85013c5cd5b86854de200b17b287.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.362ex; margin-bottom: -0.31ex; width:23.642ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}s_{c}&=e\,\alpha ^{c\,i}\\s_{c+1}&=e\,\alpha ^{(c+1)\,i}=\alpha ^{i}s_{c}\end{aligned}}}" loading="lazy"></dd></dl>

so together they allow us to calculate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

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</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> and provide some information about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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If there are two or more errors,

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<annotation encoding="application/x-tex">{\displaystyle E(x)=e_{1}x^{i_{1}}+e_{2}x^{i_{2}}+\cdots \,}</annotation>

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</math><img src="./cc6743ec395d82a065410941e003e0fdd0e38864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.001ex; height:3.176ex;" alt="{\displaystyle E(x)=e_{1}x^{i_{1}}+e_{2}x^{i_{2}}+\cdots \,}" loading="lazy"></dd></dl>

It is not immediately obvious how to begin solving the resulting syndromes for the unknowns <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{k}}">

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</math><img src="./1a24f4c64db82ae2f1e89d9861304aec8598c7e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.172ex; height:2.009ex;" alt="{\displaystyle e_{k}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{k}.}">

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</math><img src="./60baffb02c3b1f0d9eb09027b9dc76d5a40fd340.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.538ex; height:2.509ex;" alt="{\displaystyle i_{k}.}" loading="lazy">

The first step is finding, compatible with computed syndromes and with minimal possible <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t,}">

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<annotation encoding="application/x-tex">{\displaystyle t,}</annotation>

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</math><img src="./4ea3ad87830a1055c7b85c04cf940cfd3b847ae6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.486ex; height:2.343ex;" alt="{\displaystyle t,}" loading="lazy"> locator polynomial:

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)=\prod _{j=1}^{t}\left(x\alpha ^{i_{j}}-1\right)}</annotation>

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</math><img src="./dd5e6c01ee2349eb9f7a4c73fb18eeb36a709a1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:21.671ex; height:7.509ex;" alt="{\displaystyle \Lambda (x)=\prod _{j=1}^{t}\left(x\alpha ^{i_{j}}-1\right)}" loading="lazy"></dd></dl>

Three popular algorithms for this task are:

<ol><li><a href="#Peterson–Gorenstein–Zierler_algorithm">Peterson–Gorenstein–Zierler algorithm</a></li>

<li><a href="Berlekamp%E2%80%93Massey_algorithm" title="Berlekamp–Massey algorithm">Berlekamp–Massey algorithm</a></li>

<li><a href="Reed%E2%80%93Solomon_error_correction#Euclidean_decoder" title="Reed–Solomon error correction">Sugiyama Euclidean algorithm</a></li></ol>

<div class="mw-heading mw-heading4"><h4 id="Peterson–Gorenstein–Zierler_algorithm">Peterson–Gorenstein–Zierler algorithm</h4></div>

<a href="Peterson's_algorithm" title="Peterson's algorithm">Peterson's algorithm</a> is the step 2 of the generalized BCH decoding procedure. Peterson's algorithm is used to calculate the error locator polynomial coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1},\lambda _{2},\dots ,\lambda _{v}}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{1},\lambda _{2},\dots ,\lambda _{v}}</annotation>

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</math><img src="./37e7d5a53b833f33474ce174def47533d2680dc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.416ex; height:2.509ex;" alt="{\displaystyle \lambda _{1},\lambda _{2},\dots ,\lambda _{v}}" loading="lazy"> of a polynomial

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)=1+\lambda _{1}x+\lambda _{2}x^{2}+\cdots +\lambda _{v}x^{v}.}</annotation>

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</math><img src="./885f929e1a92d5dd40e775b10deb73bf9d961f33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.021ex; height:3.176ex;" alt="{\displaystyle \Lambda (x)=1+\lambda _{1}x+\lambda _{2}x^{2}+\cdots +\lambda _{v}x^{v}.}" loading="lazy"></dd></dl>

Now the procedure of the Peterson–Gorenstein–Zierler algorithm.<a href="#cite_note-8">[8]</a> Expect we have at least 2t syndromes sc, …, sc+2t−1. Let v = t.

<div><ol><li>Start by generating the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{v\times v}}">

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<annotation encoding="application/x-tex">{\displaystyle S_{v\times v}={\begin{bmatrix}s_{c}&s_{c+1}&\dots &s_{c+v-1}\\s_{c+1}&s_{c+2}&\dots &s_{c+v}\\\vdots &\vdots &\ddots &\vdots \\s_{c+v-1}&s_{c+v}&\dots &s_{c+2v-2}\end{bmatrix}}.}</annotation>

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</math><img src="./a63e2943f9af195e17edf027ae612bfd53a9e8e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:39.454ex; height:14.176ex;" alt="{\displaystyle S_{v\times v}={\begin{bmatrix}s_{c}&s_{c+1}&\dots &s_{c+v-1}\\s_{c+1}&s_{c+2}&\dots &s_{c+v}\\\vdots &\vdots &\ddots &\vdots \\s_{c+v-1}&s_{c+v}&\dots &s_{c+2v-2}\end{bmatrix}}.}" loading="lazy"></dd></dl></li><li>Generate a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{v\times 1}}">

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<annotation encoding="application/x-tex">{\displaystyle c_{v\times 1}}</annotation>

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</math><img src="./b440492de6beb8789d342b2b4ca81ec8efa3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.137ex; height:2.009ex;" alt="{\displaystyle c_{v\times 1}}" loading="lazy"> vector with elements

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<annotation encoding="application/x-tex">{\displaystyle C_{v\times 1}={\begin{bmatrix}s_{c+v}\\s_{c+v+1}\\\vdots \\s_{c+2v-1}\end{bmatrix}}.}</annotation>

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</math><img src="./3c809d59bafcf61def5e7f67704ddaef10eff082.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:19.422ex; height:13.843ex;" alt="{\displaystyle C_{v\times 1}={\begin{bmatrix}s_{c+v}\\s_{c+v+1}\\\vdots \\s_{c+2v-1}\end{bmatrix}}.}" loading="lazy"></dd></li><li>Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda _{v\times 1}={\begin{bmatrix}\lambda _{v}\\\lambda _{v-1}\\\vdots \\\lambda _{1}\end{bmatrix}}.}</annotation>

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</math><img src="./09a8002dd80a623b02c5a852156c4712df4bd81b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:16.826ex; height:13.843ex;" alt="{\displaystyle \Lambda _{v\times 1}={\begin{bmatrix}\lambda _{v}\\\lambda _{v-1}\\\vdots \\\lambda _{1}\end{bmatrix}}.}" loading="lazy"></dd></li><li>Form the matrix equation

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<annotation encoding="application/x-tex">{\displaystyle S_{v\times v}\Lambda _{v\times 1}=-C_{v\times 1\,}.}</annotation>

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</math><img src="./afa4d2c9fe794d7dfaa5280f35605b18ab88b6d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.006ex; height:2.509ex;" alt="{\displaystyle S_{v\times v}\Lambda _{v\times 1}=-C_{v\times 1\,}.}" loading="lazy"></dd></li><li>If the determinant of matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{v\times v}}">

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</math><img src="./3dff0463f5866d8c05b89b6deb1ee3d851fcbd7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.53ex; height:2.509ex;" alt="{\displaystyle S_{v\times v}}" loading="lazy"> is nonzero, then we can actually find an inverse of this matrix and solve for the values of unknown <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

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</math><img src="./0ac0a4a98a414e3480335f9ba652d12571ec6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" loading="lazy"> values.</li><li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det \left(S_{v\times v}\right)=0,}">

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<annotation encoding="application/x-tex">{\displaystyle \det \left(S_{v\times v}\right)=0,}</annotation>

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</math><img src="./11dceef961d25ee9de575a4572a15adafef39eb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.864ex; height:2.843ex;" alt="{\displaystyle \det \left(S_{v\times v}\right)=0,}" loading="lazy"> then follow

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<annotation encoding="application/x-tex">{\displaystyle v=0}</annotation>

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</math><img src="./ba3d414a23bf4ecfa36cdd039241efc60a5bd9e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.389ex; height:2.176ex;" alt="{\displaystyle v=0}" loading="lazy">

then

declare an empty error locator polynomial

stop Peterson procedure.

end

set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\leftarrow v-1}">

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</pre>

continue from the beginning of Peterson's decoding by making smaller <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{v\times v}}">

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</math><img src="./3dff0463f5866d8c05b89b6deb1ee3d851fcbd7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.53ex; height:2.509ex;" alt="{\displaystyle S_{v\times v}}" loading="lazy"></li><li>After you have values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

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<div class="mw-heading mw-heading3"><h3 id="Factor_error_locator_polynomial">Factor error locator polynomial</h3></div>

Now that you have the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (x)}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)}</annotation>

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</math><img src="./492c958ae36a2bd8fced1b0da90bb6f551fe5bf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.752ex; height:2.843ex;" alt="{\displaystyle \Lambda (x)}" loading="lazy"> polynomial, its roots can be found in the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (x)=\left(\alpha ^{i_{1}}x-1\right)\left(\alpha ^{i_{2}}x-1\right)\cdots \left(\alpha ^{i_{v}}x-1\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)=\left(\alpha ^{i_{1}}x-1\right)\left(\alpha ^{i_{2}}x-1\right)\cdots \left(\alpha ^{i_{v}}x-1\right)}</annotation>

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</math><img src="./eacde3ac7b6e03da8e1ba1105c11bcf60aa401a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.459ex; height:3.343ex;" alt="{\displaystyle \Lambda (x)=\left(\alpha ^{i_{1}}x-1\right)\left(\alpha ^{i_{2}}x-1\right)\cdots \left(\alpha ^{i_{v}}x-1\right)}" loading="lazy"> by brute force for example using the <a href="Chien_search" title="Chien search">Chien search</a> algorithm. The exponential

powers of the primitive element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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The zeros of Λ(x) are α−i1, …, α−iv.

<div class="mw-heading mw-heading3"><h3 id="Calculate_error_values">Calculate error values</h3></div>

Once the error locations are known, the next step is to determine the error values at those locations. The error values are then used to correct the received values at those locations to recover the original codeword.

For the case of binary BCH, (with all characters readable) this is trivial; just flip the bits for the received word at these positions, and we have the corrected code word. In the more general case, the error weights <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{j}}">

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</math><img src="./64afd228cf00e5024b9cdd277462d24ab97b6d3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.993ex; height:2.343ex;" alt="{\displaystyle e_{j}}" loading="lazy"> can be determined by solving the linear system

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s_{c}&=e_{1}\alpha ^{c\,i_{1}}+e_{2}\alpha ^{c\,i_{2}}+\cdots \\s_{c+1}&=e_{1}\alpha ^{(c+1)\,i_{1}}+e_{2}\alpha ^{(c+1)\,i_{2}}+\cdots \\&{}\ \vdots \end{aligned}}}</annotation>

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</math><img src="./67356e9887494e9b708024bca1e9a415c6d915a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:35.86ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}s_{c}&=e_{1}\alpha ^{c\,i_{1}}+e_{2}\alpha ^{c\,i_{2}}+\cdots \\s_{c+1}&=e_{1}\alpha ^{(c+1)\,i_{1}}+e_{2}\alpha ^{(c+1)\,i_{2}}+\cdots \\&{}\ \vdots \end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Forney_algorithm">Forney algorithm</h4></div>

However, there is a more efficient method known as the <a href="Forney_algorithm" title="Forney algorithm">Forney algorithm</a>.

Let

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<annotation encoding="application/x-tex">{\displaystyle S(x)=s_{c}+s_{c+1}x+s_{c+2}x^{2}+\cdots +s_{c+d-2}x^{d-2}.}</annotation>

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</math><img src="./45ce1f0cb1d19d5a96165851788daf372c8f34a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.283ex; height:3.176ex;" alt="{\displaystyle S(x)=s_{c}+s_{c+1}x+s_{c+2}x^{2}+\cdots +s_{c+d-2}x^{d-2}.}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle v\leqslant d-1,\lambda _{0}\neq 0\qquad \Lambda (x)=\sum _{i=0}^{v}\lambda _{i}x^{i}=\lambda _{0}\prod _{k=0}^{v}\left(\alpha ^{-i_{k}}x-1\right).}</annotation>

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</math><img src="./91cc5ad7a136f057672f918ff70f4076560be342.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.844ex; height:7.009ex;" alt="{\displaystyle v\leqslant d-1,\lambda _{0}\neq 0\qquad \Lambda (x)=\sum _{i=0}^{v}\lambda _{i}x^{i}=\lambda _{0}\prod _{k=0}^{v}\left(\alpha ^{-i_{k}}x-1\right).}" loading="lazy"></dd></dl>

And the error evaluator polynomial<a href="#cite_note-Gill-Forney-9">[9]</a>

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<annotation encoding="application/x-tex">{\displaystyle \Omega (x)\equiv S(x)\Lambda (x){\bmod {x^{d-1}}}}</annotation>

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</math><img src="./a152d042f121b3ff8e34a1e6c9d667d332f7f54a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.509ex; height:3.176ex;" alt="{\displaystyle \Omega (x)\equiv S(x)\Lambda (x){\bmod {x^{d-1}}}}" loading="lazy"></dd></dl>

Finally:

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<annotation encoding="application/x-tex">{\displaystyle \Lambda '(x)=\sum _{i=1}^{v}i\cdot \lambda _{i}x^{i-1},}</annotation>

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</math><img src="./2408086cf00891b52be1665f4d09c2fa4617ca92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.79ex; height:6.843ex;" alt="{\displaystyle \Lambda '(x)=\sum _{i=1}^{v}i\cdot \lambda _{i}x^{i-1},}" loading="lazy"></dd></dl>

where

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<annotation encoding="application/x-tex">{\displaystyle i\cdot x:=\sum _{k=1}^{i}x.}</annotation>

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</math><img src="./35d81ec5a90c1f551f73b28e5d9d59e73e025447.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.275ex; height:7.176ex;" alt="{\displaystyle i\cdot x:=\sum _{k=1}^{i}x.}" loading="lazy"></dd></dl>

Than if syndromes could be explained by an error word, which could be nonzero only on positions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{k}}">

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</math><img src="./10f93c0e4d8055ef78a64abbc9a8f4d987937aba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.891ex; height:2.509ex;" alt="{\displaystyle i_{k}}" loading="lazy">, then error values are

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<annotation encoding="application/x-tex">{\displaystyle e_{k}=-{\alpha ^{i_{k}}\Omega \left(\alpha ^{-i_{k}}\right) \over \alpha ^{c\cdot i_{k}}\Lambda '\left(\alpha ^{-i_{k}}\right)}.}</annotation>

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</math><img src="./1441699cb832cd430c4783d5cc3135cfeca373c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.798ex; height:6.843ex;" alt="{\displaystyle e_{k}=-{\alpha ^{i_{k}}\Omega \left(\alpha ^{-i_{k}}\right) \over \alpha ^{c\cdot i_{k}}\Lambda '\left(\alpha ^{-i_{k}}\right)}.}" loading="lazy"></dd></dl>

For narrow-sense BCH codes, c = 1, so the expression simplifies to:

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<annotation encoding="application/x-tex">{\displaystyle e_{k}=-{\Omega \left(\alpha ^{-i_{k}}\right) \over \Lambda '\left(\alpha ^{-i_{k}}\right)}.}</annotation>

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</math><img src="./1a889e3565f288523d402483865b7edbdc75efa0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.481ex; height:6.843ex;" alt="{\displaystyle e_{k}=-{\Omega \left(\alpha ^{-i_{k}}\right) \over \Lambda '\left(\alpha ^{-i_{k}}\right)}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Explanation_of_Forney_algorithm_computation">Explanation of Forney algorithm computation</h4></div>

It is based on <a href="Lagrange_polynomial" title="Lagrange polynomial">Lagrange interpolation</a> and techniques of <a href="Generating_function" title="Generating function">generating functions</a>.

Consider <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(x)\Lambda (x),}">

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</math><img src="./2b202aa0a430e7517a19f0722a23978ae779ab1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.037ex; height:2.843ex;" alt="{\displaystyle S(x)\Lambda (x),}" loading="lazy"> and for the sake of simplicity suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{k}=0}">

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</math><img src="./b71b48e1c6af649e1a1c0a4ebbb91474bc57bb6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.084ex; height:2.509ex;" alt="{\displaystyle k>v,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{k}=0}">

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<annotation encoding="application/x-tex">{\displaystyle S(x)\Lambda (x)=\sum _{j=0}^{\infty }\sum _{i=0}^{j}s_{j-i+1}\lambda _{i}x^{j}.}</annotation>

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</math><img src="./155ce3fd34602e5b91be08b8d416b613b2a236f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.961ex; height:7.676ex;" alt="{\displaystyle S(x)\Lambda (x)=\sum _{j=0}^{\infty }\sum _{i=0}^{j}s_{j-i+1}\lambda _{i}x^{j}.}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S(x)\Lambda (x)&=S(x)\left\{\lambda _{0}\prod _{\ell =1}^{v}\left(\alpha ^{i_{\ell }}x-1\right)\right\}\\&=\left\{\sum _{i=0}^{d-2}\sum _{j=1}^{v}e_{j}\alpha ^{(c+i)\cdot i_{j}}x^{i}\right\}\left\{\lambda _{0}\prod _{\ell =1}^{v}\left(\alpha ^{i_{\ell }}x-1\right)\right\}\\&=\left\{\sum _{j=1}^{v}e_{j}\alpha ^{ci_{j}}\sum _{i=0}^{d-2}\left(\alpha ^{i_{j}}\right)^{i}x^{i}\right\}\left\{\lambda _{0}\prod _{\ell =1}^{v}\left(\alpha ^{i_{\ell }}x-1\right)\right\}\\&=\left\{\sum _{j=1}^{v}e_{j}\alpha ^{ci_{j}}{\frac {\left(x\alpha ^{i_{j}}\right)^{d-1}-1}{x\alpha ^{i_{j}}-1}}\right\}\left\{\lambda _{0}\prod _{\ell =1}^{v}\left(\alpha ^{i_{\ell }}x-1\right)\right\}\\&=\lambda _{0}\sum _{j=1}^{v}e_{j}\alpha ^{ci_{j}}{\frac {\left(x\alpha ^{i_{j}}\right)^{d-1}-1}{x\alpha ^{i_{j}}-1}}\prod _{\ell =1}^{v}\left(\alpha ^{i_{\ell }}x-1\right)\\&=\lambda _{0}\sum _{j=1}^{v}e_{j}\alpha ^{ci_{j}}\left(\left(x\alpha ^{i_{j}}\right)^{d-1}-1\right)\prod _{\ell \in \{1,\cdots ,v\}\setminus \{j\}}\left(\alpha ^{i_{\ell }}x-1\right)\end{aligned}}}</annotation>

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</math><img src="./d0f340534f346addfa62285da5a7dbddf62446e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -23.338ex; width:64.578ex; height:47.843ex;" alt="{\displaystyle {\begin{aligned}S(x)\Lambda (x)&=S(x)\left\{\lambda _{0}\prod _{\ell =1}^{v}\left(\alpha ^{i_{\ell }}x-1\right)\right\}\\&=\left\{\sum _{i=0}^{d-2}\sum _{j=1}^{v}e_{j}\alpha ^{(c+i)\cdot i_{j}}x^{i}\right\}\left\{\lambda _{0}\prod _{\ell =1}^{v}\left(\alpha ^{i_{\ell }}x-1\right)\right\}\\&=\left\{\sum _{j=1}^{v}e_{j}\alpha ^{ci_{j}}\sum _{i=0}^{d-2}\left(\alpha ^{i_{j}}\right)^{i}x^{i}\right\}\left\{\lambda _{0}\prod _{\ell =1}^{v}\left(\alpha ^{i_{\ell }}x-1\right)\right\}\\&=\left\{\sum _{j=1}^{v}e_{j}\alpha ^{ci_{j}}{\frac {\left(x\alpha ^{i_{j}}\right)^{d-1}-1}{x\alpha ^{i_{j}}-1}}\right\}\left\{\lambda _{0}\prod _{\ell =1}^{v}\left(\alpha ^{i_{\ell }}x-1\right)\right\}\\&=\lambda _{0}\sum _{j=1}^{v}e_{j}\alpha ^{ci_{j}}{\frac {\left(x\alpha ^{i_{j}}\right)^{d-1}-1}{x\alpha ^{i_{j}}-1}}\prod _{\ell =1}^{v}\left(\alpha ^{i_{\ell }}x-1\right)\\&=\lambda _{0}\sum _{j=1}^{v}e_{j}\alpha ^{ci_{j}}\left(\left(x\alpha ^{i_{j}}\right)^{d-1}-1\right)\prod _{\ell \in \{1,\cdots ,v\}\setminus \{j\}}\left(\alpha ^{i_{\ell }}x-1\right)\end{aligned}}}" loading="lazy"></dd></dl>

We want to compute unknowns <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{j},}">

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<annotation encoding="application/x-tex">{\displaystyle e_{j},}</annotation>

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</math><img src="./60fdf5e81bd311dd236fb2b46e4a5ec524b427b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.64ex; height:2.343ex;" alt="{\displaystyle e_{j},}" loading="lazy"> and we could simplify the context by removing the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x\alpha ^{i_{j}}\right)^{d-1}}">

<msup>

<mrow>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \left(x\alpha ^{i_{j}}\right)^{d-1}}</annotation>

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</math><img src="./602bb5801ba07f5dec29c345942fcdd81b64d0dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.654ex; height:3.843ex;" alt="{\displaystyle \left(x\alpha ^{i_{j}}\right)^{d-1}}" loading="lazy"> terms. This leads to the error evaluator polynomial

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<annotation encoding="application/x-tex">{\displaystyle \Omega (x)\equiv S(x)\Lambda (x){\bmod {x^{d-1}}}.}</annotation>

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</math><img src="./158d5d77dc10f536d7f84ad00e00b7aeabfc4b9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.156ex; height:3.176ex;" alt="{\displaystyle \Omega (x)\equiv S(x)\Lambda (x){\bmod {x^{d-1}}}.}" loading="lazy"></dd></dl>

Thanks to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\leqslant d-1}">

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<annotation encoding="application/x-tex">{\displaystyle v\leqslant d-1}</annotation>

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</math><img src="./2d495fbcdcaa6b0e9c617aef7b7d5c9626eda1ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.445ex; height:2.343ex;" alt="{\displaystyle v\leqslant d-1}" loading="lazy"> we have

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<msup>

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<mrow>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \Omega (x)=-\lambda _{0}\sum _{j=1}^{v}e_{j}\alpha ^{ci_{j}}\prod _{\ell \in \{1,\cdots ,v\}\setminus \{j\}}\left(\alpha ^{i_{\ell }}x-1\right).}</annotation>

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</math><img src="./56d842e9a2dbc367a55d75d06ef7bc08f1fca98e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:45.277ex; height:7.343ex;" alt="{\displaystyle \Omega (x)=-\lambda _{0}\sum _{j=1}^{v}e_{j}\alpha ^{ci_{j}}\prod _{\ell \in \{1,\cdots ,v\}\setminus \{j\}}\left(\alpha ^{i_{\ell }}x-1\right).}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation>

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</math><img src="./0ac0a4a98a414e3480335f9ba652d12571ec6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" loading="lazy"> (the Lagrange interpolation trick) the sum degenerates to only one summand for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\alpha ^{-i_{k}}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle x=\alpha ^{-i_{k}}}</annotation>

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</math><img src="./e4543fbd413fe4af4131745612ba9aa8224b5931.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.854ex; height:2.676ex;" alt="{\displaystyle x=\alpha ^{-i_{k}}}" loading="lazy">

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<msup>

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<mrow>

<mrow>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \Omega \left(\alpha ^{-i_{k}}\right)=-\lambda _{0}e_{k}\alpha ^{c\cdot i_{k}}\prod _{\ell \in \{1,\cdots ,v\}\setminus \{k\}}\left(\alpha ^{i_{\ell }}\alpha ^{-i_{k}}-1\right).}</annotation>

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</math><img src="./eefca3d0ab47199b4a72cbfe5e4129a1ac9bdf29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:49.007ex; height:6.009ex;" alt="{\displaystyle \Omega \left(\alpha ^{-i_{k}}\right)=-\lambda _{0}e_{k}\alpha ^{c\cdot i_{k}}\prod _{\ell \in \{1,\cdots ,v\}\setminus \{k\}}\left(\alpha ^{i_{\ell }}\alpha ^{-i_{k}}-1\right).}" loading="lazy"></dd></dl>

To get <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{k}}</annotation>

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</math><img src="./1a24f4c64db82ae2f1e89d9861304aec8598c7e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.172ex; height:2.009ex;" alt="{\displaystyle e_{k}}" loading="lazy"> we just should get rid of the product. We could compute the product directly from already computed roots <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{-i_{j}}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha ^{-i_{j}}}</annotation>

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</math><img src="./b7dda7ff4dd51c9b0ec9e97f81de0562b492bd73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.28ex; height:2.676ex;" alt="{\displaystyle \alpha ^{-i_{j}}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ,}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda ,}</annotation>

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</math><img src="./c2689167a7bad3bc1d54314e9870122258c1028f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.26ex; height:2.509ex;" alt="{\displaystyle \Lambda ,}" loading="lazy"> but we could use simpler form.

As <a href="Formal_derivative" title="Formal derivative">formal derivative</a>

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<annotation encoding="application/x-tex">{\displaystyle \Lambda '(x)=\lambda _{0}\sum _{j=1}^{v}\alpha ^{i_{j}}\prod _{\ell \in \{1,\cdots ,v\}\setminus \{j\}}\left(\alpha ^{i_{\ell }}x-1\right),}</annotation>

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</math><img src="./af98fab10676e8490205eded635f678632c750bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:41.383ex; height:7.343ex;" alt="{\displaystyle \Lambda '(x)=\lambda _{0}\sum _{j=1}^{v}\alpha ^{i_{j}}\prod _{\ell \in \{1,\cdots ,v\}\setminus \{j\}}\left(\alpha ^{i_{\ell }}x-1\right),}" loading="lazy"></dd></dl>

we get again only one summand in

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \Lambda '\left(\alpha ^{-i_{k}}\right)=\lambda _{0}\alpha ^{i_{k}}\prod _{\ell \in \{1,\cdots ,v\}\setminus \{k\}}\left(\alpha ^{i_{\ell }}\alpha ^{-i_{k}}-1\right).}</annotation>

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</math><img src="./f3178682b1427ebfdac05d6b84c9c3b5067a61ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:44.477ex; height:6.009ex;" alt="{\displaystyle \Lambda '\left(\alpha ^{-i_{k}}\right)=\lambda _{0}\alpha ^{i_{k}}\prod _{\ell \in \{1,\cdots ,v\}\setminus \{k\}}\left(\alpha ^{i_{\ell }}\alpha ^{-i_{k}}-1\right).}" loading="lazy"></dd></dl>

So finally

<msub>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle e_{k}=-{\frac {\alpha ^{i_{k}}\Omega \left(\alpha ^{-i_{k}}\right)}{\alpha ^{c\cdot i_{k}}\Lambda '\left(\alpha ^{-i_{k}}\right)}}.}</annotation>

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</math><img src="./b744e09ef8fc64e7510ff721f622a696819b2885.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.798ex; height:6.843ex;" alt="{\displaystyle e_{k}=-{\frac {\alpha ^{i_{k}}\Omega \left(\alpha ^{-i_{k}}\right)}{\alpha ^{c\cdot i_{k}}\Lambda '\left(\alpha ^{-i_{k}}\right)}}.}" loading="lazy"></dd></dl>

This formula is advantageous when one computes the formal derivative of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)=\sum _{i=1}^{v}\lambda _{i}x^{i}}</annotation>

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</math><img src="./cfd4bdfbb76ab19df3d33227a3b719acb17b242a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.877ex; height:6.843ex;" alt="{\displaystyle \Lambda (x)=\sum _{i=1}^{v}\lambda _{i}x^{i}}" loading="lazy"></dd></dl>

yielding:

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<annotation encoding="application/x-tex">{\displaystyle \Lambda '(x)=\sum _{i=1}^{v}i\cdot \lambda _{i}x^{i-1},}</annotation>

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where

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<annotation encoding="application/x-tex">{\displaystyle i\cdot x:=\sum _{k=1}^{i}x.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Decoding_based_on_extended_Euclidean_algorithm">Decoding based on extended Euclidean algorithm</h3></div>

An alternate process of finding both the polynomial Λ and the error locator polynomial is based on Yasuo Sugiyama's adaptation of the <a href="Extended_Euclidean_algorithm" title="Extended Euclidean algorithm">Extended Euclidean algorithm</a>.<a href="#cite_note-10">[10]</a> Correction of unreadable characters could be incorporated to the algorithm easily as well.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1},...,k_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle k_{1},...,k_{k}}</annotation>

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</math><img src="./91daeb8eaa7ae2003bbc65d1fb01ff3dd213f5ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.735ex; height:2.509ex;" alt="{\displaystyle k_{1},...,k_{k}}" loading="lazy"> be positions of unreadable characters. One creates polynomial localising these positions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (x)=\prod _{i=1}^{k}\left(x\alpha ^{k_{i}}-1\right).}">

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</munderover>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (x)=\prod _{i=1}^{k}\left(x\alpha ^{k_{i}}-1\right).}</annotation>

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</math><img src="./bce723e3909ca7302e4bb3b0b69dcf40670402eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.744ex; height:7.343ex;" alt="{\displaystyle \Gamma (x)=\prod _{i=1}^{k}\left(x\alpha ^{k_{i}}-1\right).}" loading="lazy">

Set values on unreadable positions to 0 and compute the syndromes.

As we have already defined for the Forney formula let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(x)=\sum _{i=0}^{d-2}s_{c+i}x^{i}.}">

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<annotation encoding="application/x-tex">{\displaystyle S(x)=\sum _{i=0}^{d-2}s_{c+i}x^{i}.}</annotation>

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</math><img src="./e07a851cfd06a507f57c79af30674b1aa20107fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.136ex; height:7.343ex;" alt="{\displaystyle S(x)=\sum _{i=0}^{d-2}s_{c+i}x^{i}.}" loading="lazy">

Let us run extended Euclidean algorithm for locating least common divisor of polynomials <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(x)\Gamma (x)}">

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<annotation encoding="application/x-tex">{\displaystyle S(x)\Gamma (x)}</annotation>

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</math><img src="./d881d2fbb20822b4b8b5740f79daf1e7b3702cc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.23ex; height:2.843ex;" alt="{\displaystyle S(x)\Gamma (x)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{d-1}.}">

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<annotation encoding="application/x-tex">{\displaystyle x^{d-1}.}</annotation>

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</math><img src="./b4098f0f2fb2deabc075b76d1fe8989be2f680a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.169ex; height:2.676ex;" alt="{\displaystyle x^{d-1}.}" loading="lazy">

The goal is not to find the least common divisor, but a polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(x)}">

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<annotation encoding="application/x-tex">{\displaystyle r(x)}</annotation>

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</math><img src="./0d457e88abd861f46ede37d27c4669c747367109.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.188ex; height:2.843ex;" alt="{\displaystyle r(x)}" loading="lazy"> of degree at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor (d+k-3)/2\rfloor }">

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<annotation encoding="application/x-tex">{\displaystyle \lfloor (d+k-3)/2\rfloor }</annotation>

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</math><img src="./d23f7603ce8b5270bd520e51a5e88a1b10bf0ae1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.469ex; height:2.843ex;" alt="{\displaystyle \lfloor (d+k-3)/2\rfloor }" loading="lazy"> and polynomials <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(x),b(x)}">

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<annotation encoding="application/x-tex">{\displaystyle a(x),b(x)}</annotation>

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</math><img src="./6c8660dba3ec310d9bc1f307ffd04fe31f66fb92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.539ex; height:2.843ex;" alt="{\displaystyle a(x),b(x)}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(x)=a(x)S(x)\Gamma (x)+b(x)x^{d-1}.}">

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<annotation encoding="application/x-tex">{\displaystyle r(x)=a(x)S(x)\Gamma (x)+b(x)x^{d-1}.}</annotation>

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</math><img src="./90d40d9e67418c9f24ac7ce7f0046b2dcf52a9c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.031ex; height:3.176ex;" alt="{\displaystyle r(x)=a(x)S(x)\Gamma (x)+b(x)x^{d-1}.}" loading="lazy">

Low degree of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(x)}">

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<annotation encoding="application/x-tex">{\displaystyle r(x)}</annotation>

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</math><img src="./0d457e88abd861f46ede37d27c4669c747367109.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.188ex; height:2.843ex;" alt="{\displaystyle r(x)}" loading="lazy"> guarantees, that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(x)}">

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<annotation encoding="application/x-tex">{\displaystyle a(x)}</annotation>

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</math><img src="./5d685172ca5d7f10d1e9dfd16cffc5a54bf1301a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.369ex; height:2.843ex;" alt="{\displaystyle a(x)}" loading="lazy"> would satisfy extended (by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

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</math><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy">) defining conditions for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda .}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda .}</annotation>

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Defining <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Xi (x)=a(x)\Gamma (x)}">

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<annotation encoding="application/x-tex">{\displaystyle \Xi (x)=a(x)\Gamma (x)}</annotation>

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</math><img src="./34cd859a1bb215efce99100fc075b8fe7e176af7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.748ex; height:2.843ex;" alt="{\displaystyle \Xi (x)=a(x)\Gamma (x)}" loading="lazy"> and using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Xi }">

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<annotation encoding="application/x-tex">{\displaystyle \Xi }</annotation>

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</math><img src="./6fcfdbcd1348cf9e34618a31dbdcb36361406220.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \Xi }" loading="lazy"> on the place of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (x)}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)}</annotation>

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The main advantage of the algorithm is that it meanwhile computes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega (x)=S(x)\Xi (x){\bmod {x}}^{d-1}=r(x)}">

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<annotation encoding="application/x-tex">{\displaystyle \Omega (x)=S(x)\Xi (x){\bmod {x}}^{d-1}=r(x)}</annotation>

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</math><img src="./46d040c641314b79776ba4a5db2d2b3490efcd72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.733ex; height:3.176ex;" alt="{\displaystyle \Omega (x)=S(x)\Xi (x){\bmod {x}}^{d-1}=r(x)}" loading="lazy"> required in the Forney formula.

<div class="mw-heading mw-heading4"><h4 id="Explanation_of_the_decoding_process">Explanation of the decoding process</h4></div>

The goal is to find a codeword which differs from the received word minimally as possible on readable positions. When expressing the received word as a sum of nearest codeword and error word, we are trying to find error word with minimal number of non-zeros on readable positions. Syndrom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle s_{i}}</annotation>

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</math><img src="./cfda82668232cbdc0874ed28ab8b6079420d1ffe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.009ex;" alt="{\displaystyle s_{i}}" loading="lazy"> restricts error word by condition

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<annotation encoding="application/x-tex">{\displaystyle s_{i}=\sum _{j=0}^{n-1}e_{j}\alpha ^{ij}.}</annotation>

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</math><img src="./7195725f03201300c4b70c3d2fc3b63bc130ff76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.335ex; height:7.676ex;" alt="{\displaystyle s_{i}=\sum _{j=0}^{n-1}e_{j}\alpha ^{ij}.}" loading="lazy"></dd></dl>

We could write these conditions separately or we could create polynomial

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<annotation encoding="application/x-tex">{\displaystyle S(x)=\sum _{i=0}^{d-2}s_{c+i}x^{i}}</annotation>

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</math><img src="./c17124721739e23aed482a638f69da794cddea09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.489ex; height:7.343ex;" alt="{\displaystyle S(x)=\sum _{i=0}^{d-2}s_{c+i}x^{i}}" loading="lazy"></dd></dl>

and compare coefficients near powers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d-2.}">

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<annotation encoding="application/x-tex">{\displaystyle d-2.}</annotation>

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</math><img src="./6e98d003a8996dcf27e73c75ac97c1eb1ec8c032.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.866ex; height:2.343ex;" alt="{\displaystyle d-2.}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle S(x){\stackrel {\{0,\cdots ,\,d-2\}}{=}}E(x)=\sum _{i=0}^{d-2}\sum _{j=0}^{n-1}e_{j}\alpha ^{ij}\alpha ^{cj}x^{i}.}</annotation>

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</math><img src="./dd434344027ca6ccf230f76996552415080e5d92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:39.633ex; height:7.676ex;" alt="{\displaystyle S(x){\stackrel {\{0,\cdots ,\,d-2\}}{=}}E(x)=\sum _{i=0}^{d-2}\sum _{j=0}^{n-1}e_{j}\alpha ^{ij}\alpha ^{cj}x^{i}.}" loading="lazy"></dd></dl>

Suppose there is unreadable letter on position <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1},}">

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<annotation encoding="application/x-tex">{\displaystyle k_{1},}</annotation>

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</math><img src="./ca3fde9524e03d29f6cadcdf608f171943f0ce02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.912ex; height:2.509ex;" alt="{\displaystyle k_{1},}" loading="lazy"> we could replace set of syndromes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{s_{c},\cdots ,s_{c+d-2}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{s_{c},\cdots ,s_{c+d-2}\}}</annotation>

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</math><img src="./cb399b45eb106c423c279a90f36c1a81955ebefa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.811ex; height:2.843ex;" alt="{\displaystyle \{s_{c},\cdots ,s_{c+d-2}\}}" loading="lazy"> by set of syndromes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{t_{c},\cdots ,t_{c+d-3}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{t_{c},\cdots ,t_{c+d-3}\}}</annotation>

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</math><img src="./0a21d61a328cd73fa0ac1f0d66575fdb95aa9dd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.31ex; height:2.843ex;" alt="{\displaystyle \{t_{c},\cdots ,t_{c+d-3}\}}" loading="lazy"> defined by equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}=\alpha ^{k_{1}}s_{i}-s_{i+1}.}">

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<annotation encoding="application/x-tex">{\displaystyle t_{i}=\alpha ^{k_{1}}s_{i}-s_{i+1}.}</annotation>

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</math><img src="./6c402343e77793bbe2b525988cc2e5a357547819.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.514ex; height:3.009ex;" alt="{\displaystyle t_{i}=\alpha ^{k_{1}}s_{i}-s_{i+1}.}" loading="lazy"> Suppose for an error word all restrictions by original set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{s_{c},\cdots ,s_{c+d-2}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{s_{c},\cdots ,s_{c+d-2}\}}</annotation>

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than

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<annotation encoding="application/x-tex">{\displaystyle t_{i}=\alpha ^{k_{1}}s_{i}-s_{i+1}=\alpha ^{k_{1}}\sum _{j=0}^{n-1}e_{j}\alpha ^{ij}-\sum _{j=0}^{n-1}e_{j}\alpha ^{j}\alpha ^{ij}=\sum _{j=0}^{n-1}e_{j}\left(\alpha ^{k_{1}}-\alpha ^{j}\right)\alpha ^{ij}.}</annotation>

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</math><img src="./ed575912e28aad13776a6b9f6c968f53d39cfd45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:70.393ex; height:7.676ex;" alt="{\displaystyle t_{i}=\alpha ^{k_{1}}s_{i}-s_{i+1}=\alpha ^{k_{1}}\sum _{j=0}^{n-1}e_{j}\alpha ^{ij}-\sum _{j=0}^{n-1}e_{j}\alpha ^{j}\alpha ^{ij}=\sum _{j=0}^{n-1}e_{j}\left(\alpha ^{k_{1}}-\alpha ^{j}\right)\alpha ^{ij}.}" loading="lazy"></dd></dl>

New set of syndromes restricts error vector

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<annotation encoding="application/x-tex">{\displaystyle f_{j}=e_{j}\left(\alpha ^{k_{1}}-\alpha ^{j}\right)}</annotation>

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</math><img src="./bcf0183f4ea7c1135eadac338528888c5a9a157f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.303ex; height:3.343ex;" alt="{\displaystyle f_{j}=e_{j}\left(\alpha ^{k_{1}}-\alpha ^{j}\right)}" loading="lazy"></dd></dl>

the same way the original set of syndromes restricted the error vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{j}.}">

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<annotation encoding="application/x-tex">{\displaystyle e_{j}.}</annotation>

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</math><img src="./c110c74ff60914e2c7e96136386457dfdfb05d4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.64ex; height:2.343ex;" alt="{\displaystyle e_{j}.}" loading="lazy"> Except the coordinate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1},}">

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<annotation encoding="application/x-tex">{\displaystyle k_{1},}</annotation>

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</math><img src="./ca3fde9524e03d29f6cadcdf608f171943f0ce02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.912ex; height:2.509ex;" alt="{\displaystyle k_{1},}" loading="lazy"> where we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{k_{1}}=0,}">

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<annotation encoding="application/x-tex">{\displaystyle f_{k_{1}}=0,}</annotation>

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</math><img src="./d57536532000199e72f0a96f17a7908cfd8b9289.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.967ex; height:2.843ex;" alt="{\displaystyle f_{k_{1}}=0,}" loading="lazy"> an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{j}}</annotation>

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</math><img src="./acc195ab3f9d65994b47774eb013601d09217aee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.049ex; height:2.843ex;" alt="{\displaystyle f_{j}}" loading="lazy"> is zero, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{j}=0.}">

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<annotation encoding="application/x-tex">{\displaystyle e_{j}=0.}</annotation>

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</math><img src="./e1d97c6916ad0ebe3a9be66150bf38de219e38d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.901ex; height:2.843ex;" alt="{\displaystyle e_{j}=0.}" loading="lazy"> For the goal of locating error positions we could change the set of syndromes in the similar way to reflect all unreadable characters. This shortens the set of syndromes by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k.}">

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<annotation encoding="application/x-tex">{\displaystyle k.}</annotation>

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</math><img src="./bcb6778a29f576eb23da1dbddffb73b2571359ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:2.176ex;" alt="{\displaystyle k.}" loading="lazy">

In polynomial formulation, the replacement of syndromes set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{s_{c},\cdots ,s_{c+d-2}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{s_{c},\cdots ,s_{c+d-2}\}}</annotation>

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</math><img src="./cb399b45eb106c423c279a90f36c1a81955ebefa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.811ex; height:2.843ex;" alt="{\displaystyle \{s_{c},\cdots ,s_{c+d-2}\}}" loading="lazy"> by syndromes set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{t_{c},\cdots ,t_{c+d-3}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{t_{c},\cdots ,t_{c+d-3}\}}</annotation>

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</munderover>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle T(x)=\sum _{i=0}^{d-3}t_{c+i}x^{i}=\alpha ^{k_{1}}\sum _{i=0}^{d-3}s_{c+i}x^{i}-\sum _{i=1}^{d-2}s_{c+i}x^{i-1}.}</annotation>

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</math><img src="./27d6345585f791d3e38e0305cbd0e911f3a1a37d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.36ex; height:7.343ex;" alt="{\displaystyle T(x)=\sum _{i=0}^{d-3}t_{c+i}x^{i}=\alpha ^{k_{1}}\sum _{i=0}^{d-3}s_{c+i}x^{i}-\sum _{i=1}^{d-2}s_{c+i}x^{i-1}.}" loading="lazy"></dd></dl>

Therefore,

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<annotation encoding="application/x-tex">{\displaystyle xT(x){\stackrel {\{1,\cdots ,\,d-2\}}{=}}\left(x\alpha ^{k_{1}}-1\right)S(x).}</annotation>

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</math><img src="./75794b2b4a58c9dce46f0db9534e93b5b6a2278f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.688ex; height:4.176ex;" alt="{\displaystyle xT(x){\stackrel {\{1,\cdots ,\,d-2\}}{=}}\left(x\alpha ^{k_{1}}-1\right)S(x).}" loading="lazy"></dd></dl>

After replacement of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(x)}">

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<annotation encoding="application/x-tex">{\displaystyle S(x)}</annotation>

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</math><img src="./1886b5a535ed8f168a7c3a83afc8ca440edcdc6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle S(x)}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(x)\Gamma (x)}">

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<annotation encoding="application/x-tex">{\displaystyle S(x)\Gamma (x)}</annotation>

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</math><img src="./d881d2fbb20822b4b8b5740f79daf1e7b3702cc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.23ex; height:2.843ex;" alt="{\displaystyle S(x)\Gamma (x)}" loading="lazy">, one would require equation for coefficients near powers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k,\cdots ,d-2.}">

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<annotation encoding="application/x-tex">{\displaystyle k,\cdots ,d-2.}</annotation>

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</math><img src="./920ea81f0fe983e13b46766a9d83329a9d77b6b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.255ex; height:2.509ex;" alt="{\displaystyle k,\cdots ,d-2.}" loading="lazy">

One could consider looking for error positions from the point of view of eliminating influence of given positions similarly as for unreadable characters. If we found <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> positions such that eliminating their influence leads to obtaining set of syndromes consisting of all zeros, than there exists error vector with errors only on these coordinates.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)}</annotation>

</semantics>

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(x)\Gamma (x)\Lambda (x){\stackrel {\{k+v,\cdots ,d-2\}}{=}}0.}</annotation>

</semantics>

</math><img src="./0706287d781655cd898db678aad3695e65b9b510.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.168ex; height:4.009ex;" alt="{\displaystyle S(x)\Gamma (x)\Lambda (x){\stackrel {\{k+v,\cdots ,d-2\}}{=}}0.}" loading="lazy"></dd></dl>

In Euclidean algorithm, we try to correct at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}(d-1-k)}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}(d-1-k)}</annotation>

</semantics>

</math><img src="./8549d4bdae044159fb532e7a19802853fe517d60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.738ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}(d-1-k)}" loading="lazy"> errors (on readable positions), because with bigger error count there could be more codewords in the same distance from the received word. Therefore, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)}</annotation>

</semantics>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k+\left\lfloor {\frac {1}{2}}(d-1-k)\right\rfloor .}</annotation>

</semantics>

</math><img src="./f01a2bc25df6184cb5176ecb1b5d46ddf250728a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.874ex; height:6.176ex;" alt="{\displaystyle k+\left\lfloor {\frac {1}{2}}(d-1-k)\right\rfloor .}" loading="lazy"></dd></dl>

In Forney formula, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)}</annotation>

</semantics>

It could happen that the Euclidean algorithm finds <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)}</annotation>

</semantics>

</math><img src="./492c958ae36a2bd8fced1b0da90bb6f551fe5bf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.752ex; height:2.843ex;" alt="{\displaystyle \Lambda (x)}" loading="lazy"> of degree higher than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}(d-1-k)}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}(d-1-k)}</annotation>

</semantics>

</math><img src="./8549d4bdae044159fb532e7a19802853fe517d60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.738ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}(d-1-k)}" loading="lazy"> having number of different roots equal to its degree, where the Fourney formula would be able to correct errors in all its roots, anyway correcting such many errors could be risky (especially with no other restrictions on received word). Usually after getting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)}</annotation>

</semantics>

</math><img src="./492c958ae36a2bd8fced1b0da90bb6f551fe5bf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.752ex; height:2.843ex;" alt="{\displaystyle \Lambda (x)}" loading="lazy"> of higher degree, we decide not to correct the errors. Correction could fail in the case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)}</annotation>

</semantics>

</math><img src="./492c958ae36a2bd8fced1b0da90bb6f551fe5bf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.752ex; height:2.843ex;" alt="{\displaystyle \Lambda (x)}" loading="lazy"> has roots with higher multiplicity or the number of roots is smaller than its degree. Fail could be detected as well by Forney formula returning error outside the transmitted alphabet.

<div class="mw-heading mw-heading3"><h3 id="Correct_the_errors">Correct the errors</h3></div>

Using the error values and error location, correct the errors and form a corrected code vector by subtracting error values at error locations.

<div class="mw-heading mw-heading3"><h3 id="Decoding_examples">Decoding examples</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Decoding_of_binary_code_without_unreadable_characters">Decoding of binary code without unreadable characters</h4></div>

Consider a BCH code in GF(24) with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=7}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d=7}</annotation>

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</math><img src="./8a608ac8e786da673f07f9647c4787e4bfdd4ee9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.176ex;" alt="{\displaystyle d=7}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=x^{10}+x^{8}+x^{5}+x^{4}+x^{2}+x+1}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

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</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle g(x)=x^{10}+x^{8}+x^{5}+x^{4}+x^{2}+x+1}</annotation>

</semantics>

</math><img src="./cc625dd562e527c19ca496aade0ffa5a887ac5e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.63ex; height:3.176ex;" alt="{\displaystyle g(x)=x^{10}+x^{8}+x^{5}+x^{4}+x^{2}+x+1}" loading="lazy">. (This is used in <a href="QR_code" title="QR code">QR codes</a>.) Let the message to be transmitted be [1 1 0 1 1], or in polynomial notation, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(x)=x^{4}+x^{3}+x+1.}">

<msup>

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</msup>

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle M(x)=x^{4}+x^{3}+x+1.}</annotation>

</semantics>

</math><img src="./6498069d6ceb982ab2b761f6d58e468144df0981.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.108ex; height:3.176ex;" alt="{\displaystyle M(x)=x^{4}+x^{3}+x+1.}" loading="lazy">

The "checksum" symbols are calculated by dividing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{10}M(x)}">

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x^{10}M(x)}</annotation>

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</math><img src="./e2c7ac3d65218310da8f1654c834082a39819e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.787ex; height:3.176ex;" alt="{\displaystyle x^{10}M(x)}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation>

</semantics>

</math><img src="./c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" loading="lazy"> and taking the remainder, resulting in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{9}+x^{4}+x^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x^{9}+x^{4}+x^{2}}</annotation>

</semantics>

</math><img src="./56265049b724da8f585703626bac6d530e46294c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.833ex; height:2.843ex;" alt="{\displaystyle x^{9}+x^{4}+x^{2}}" loading="lazy"> or [ 1 0 0 0 0 1 0 1 0 0 ]. These are appended to the message, so the transmitted codeword is [ 1 1 0 1 1 1 0 0 0 0 1 0 1 0 0 ].

Now, imagine that there are two bit-errors in the transmission, so the received codeword is [ 1 <style data-mw-deduplicate="TemplateStyles:r1239334494">

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</style>0 0 1 1 1 0 0 0 1 1 0 1 0 0 ]. In polynomial notation:

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle R(x)=C(x)+x^{13}+x^{5}=x^{14}+x^{11}+x^{10}+x^{9}+x^{5}+x^{4}+x^{2}}</annotation>

</semantics>

</math><img src="./5a922dc6654832ca178af49ad6fe815edb37bad9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.472ex; height:3.176ex;" alt="{\displaystyle R(x)=C(x)+x^{13}+x^{5}=x^{14}+x^{11}+x^{10}+x^{9}+x^{5}+x^{4}+x^{2}}" loading="lazy"></dd></dl>

In order to correct the errors, first calculate the syndromes. Taking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0010,}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \alpha =0010,}</annotation>

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</math><img src="./ad47436368c353bed711eaa7086fe137f0308417.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.883ex; height:2.509ex;" alt="{\displaystyle \alpha =0010,}" loading="lazy"> we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1}=R(\alpha ^{1})=1011,}">

<msub>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle s_{1}=R(\alpha ^{1})=1011,}</annotation>

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</math><img src="./27b1a673ebf960ecb7901ba2876e75a38ec764bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.753ex; height:3.176ex;" alt="{\displaystyle s_{1}=R(\alpha ^{1})=1011,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{2}=1001,}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle s_{2}=1001,}</annotation>

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</math><img src="./6945a2e3facf7cc568c45c731625cc9b661b47be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.54ex; height:2.509ex;" alt="{\displaystyle s_{2}=1001,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{3}=1011,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{3}=1011,}</annotation>

</semantics>

</math><img src="./098ca69f33a4b4b923cafb27000768ab59d86c11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.54ex; height:2.509ex;" alt="{\displaystyle s_{3}=1011,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{4}=1101,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{4}=1101,}</annotation>

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</math><img src="./75ec9d2d7add55890c288431063f000918869491.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.54ex; height:2.509ex;" alt="{\displaystyle s_{4}=1101,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{5}=0001,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{5}=0001,}</annotation>

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</math><img src="./af0d0c7c5df613725bcc597b60f3675cf54dc9ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.54ex; height:2.509ex;" alt="{\displaystyle s_{5}=0001,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{6}=1001.}">

<msub>

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</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle s_{6}=1001.}</annotation>

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</math><img src="./90add272add8c2a9d7d9d9e6ff927aa8b6ab6a8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.54ex; height:2.509ex;" alt="{\displaystyle s_{6}=1001.}" loading="lazy">

Next, apply the Peterson procedure by row-reducing the following <a href="Augmented_matrix" title="Augmented matrix">augmented matrix</a>.

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<annotation encoding="application/x-tex">{\displaystyle \left[S_{3\times 3}|C_{3\times 1}\right]={\begin{bmatrix}s_{1}&s_{2}&s_{3}&s_{4}\\s_{2}&s_{3}&s_{4}&s_{5}\\s_{3}&s_{4}&s_{5}&s_{6}\end{bmatrix}}={\begin{bmatrix}1011&1001&1011&1101\\1001&1011&1101&0001\\1011&1101&0001&1001\end{bmatrix}}\Rightarrow {\begin{bmatrix}0001&0000&1000&0111\\0000&0001&1011&0001\\0000&0000&0000&0000\end{bmatrix}}}</annotation>

</semantics>

</math><img src="./01ae26421a97007762e7db564652697943f6c6dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:99.385ex; height:9.176ex;" alt="{\displaystyle \left[S_{3\times 3}|C_{3\times 1}\right]={\begin{bmatrix}s_{1}&s_{2}&s_{3}&s_{4}\\s_{2}&s_{3}&s_{4}&s_{5}\\s_{3}&s_{4}&s_{5}&s_{6}\end{bmatrix}}={\begin{bmatrix}1011&1001&1011&1101\\1001&1011&1101&0001\\1011&1101&0001&1001\end{bmatrix}}\Rightarrow {\begin{bmatrix}0001&0000&1000&0111\\0000&0001&1011&0001\\0000&0000&0000&0000\end{bmatrix}}}" loading="lazy"></dd></dl>

Due to the zero row, S3×3 is singular, which is no surprise since only two errors were introduced into the codeword.

However, the upper-left corner of the matrix is identical to [S2×2 | C2×1], which gives rise to the solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{2}=1000,}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{2}=1000,}</annotation>

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</math><img src="./8d74a5a264ce77cedb288e535448aebcfea9c02e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.805ex; height:2.509ex;" alt="{\displaystyle \lambda _{2}=1000,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}=1011.}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{1}=1011.}</annotation>

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</math><img src="./6d29d1bacae348911ca926016bdb45b4b006929d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.805ex; height:2.509ex;" alt="{\displaystyle \lambda _{1}=1011.}" loading="lazy">

The resulting error locator polynomial is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (x)=1000x^{2}+1011x+0001,}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)=1000x^{2}+1011x+0001,}</annotation>

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</math><img src="./0126e3ae0a28bee14ed6061b10508bfc61785971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.841ex; height:3.176ex;" alt="{\displaystyle \Lambda (x)=1000x^{2}+1011x+0001,}" loading="lazy"> which has zeros at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0100=\alpha ^{-13}}">

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<annotation encoding="application/x-tex">{\displaystyle 0100=\alpha ^{-13}}</annotation>

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</math><img src="./3a1c1f04f68c6d41493e59ffd1f96367d31ee7c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.391ex; height:2.676ex;" alt="{\displaystyle 0100=\alpha ^{-13}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0111=\alpha ^{-5}.}">

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<annotation encoding="application/x-tex">{\displaystyle 0111=\alpha ^{-5}.}</annotation>

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</math><img src="./04312d7444a37ec291ec24f48f3aa78ef1f758c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.216ex; height:2.676ex;" alt="{\displaystyle 0111=\alpha ^{-5}.}" loading="lazy">

The exponents of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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There is no need to calculate the error values in this example, as the only possible value is 1.

<div class="mw-heading mw-heading4"><h4 id="Decoding_with_unreadable_characters">Decoding with unreadable characters</h4></div>

Suppose the same scenario, but the received word has two unreadable characters [ 1 0 0 ? 1 1 ? 0 0 1 1 0 1 0 0 ]. We replace the unreadable characters by zeros while creating the polynomial reflecting their positions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (x)=\left(\alpha ^{8}x-1\right)\left(\alpha ^{11}x-1\right).}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (x)=\left(\alpha ^{8}x-1\right)\left(\alpha ^{11}x-1\right).}</annotation>

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</math><img src="./5e87299f69934afa2abb8c7edd66582895879cc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.942ex; height:3.343ex;" alt="{\displaystyle \Gamma (x)=\left(\alpha ^{8}x-1\right)\left(\alpha ^{11}x-1\right).}" loading="lazy"> We compute the syndromes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1}=\alpha ^{-7},s_{2}=\alpha ^{1},s_{3}=\alpha ^{4},s_{4}=\alpha ^{2},s_{5}=\alpha ^{5},}">

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<annotation encoding="application/x-tex">{\displaystyle s_{1}=\alpha ^{-7},s_{2}=\alpha ^{1},s_{3}=\alpha ^{4},s_{4}=\alpha ^{2},s_{5}=\alpha ^{5},}</annotation>

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</math><img src="./0bf90d39f0828c1a1359176b41257da6929b7388.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:44.986ex; height:3.009ex;" alt="{\displaystyle s_{1}=\alpha ^{-7},s_{2}=\alpha ^{1},s_{3}=\alpha ^{4},s_{4}=\alpha ^{2},s_{5}=\alpha ^{5},}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{6}=\alpha ^{-7}.}">

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</math><img src="./ad956782f15f8f363d0f4e7332906bf9e3fd86be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.71ex; height:3.009ex;" alt="{\displaystyle s_{6}=\alpha ^{-7}.}" loading="lazy"> (Using log notation which is independent on GF(24) isomorphisms. For computation checking we can use the same representation for addition as was used in previous example. <a href="Hexadecimal" title="Hexadecimal">Hexadecimal</a> description of the powers of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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Let us make syndrome polynomial

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<annotation encoding="application/x-tex">{\displaystyle S(x)=\alpha ^{-7}+\alpha ^{1}x+\alpha ^{4}x^{2}+\alpha ^{2}x^{3}+\alpha ^{5}x^{4}+\alpha ^{-7}x^{5},}</annotation>

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</math><img src="./ccb38616c591490c55f1ce855e455f82acf91e4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.259ex; height:3.176ex;" alt="{\displaystyle S(x)=\alpha ^{-7}+\alpha ^{1}x+\alpha ^{4}x^{2}+\alpha ^{2}x^{3}+\alpha ^{5}x^{4}+\alpha ^{-7}x^{5},}" loading="lazy"></dd></dl>

compute

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<annotation encoding="application/x-tex">{\displaystyle S(x)\Gamma (x)=\alpha ^{-7}+\alpha ^{4}x+\alpha ^{-1}x^{2}+\alpha ^{6}x^{3}+\alpha ^{-1}x^{4}+\alpha ^{5}x^{5}+\alpha ^{7}x^{6}+\alpha ^{-3}x^{7}.}</annotation>

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</math><img src="./8f5b30ffcc2b375861070e16bd0ed75acafeb177.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:73.94ex; height:3.176ex;" alt="{\displaystyle S(x)\Gamma (x)=\alpha ^{-7}+\alpha ^{4}x+\alpha ^{-1}x^{2}+\alpha ^{6}x^{3}+\alpha ^{-1}x^{4}+\alpha ^{5}x^{5}+\alpha ^{7}x^{6}+\alpha ^{-3}x^{7}.}" loading="lazy"></dd></dl>

Run the extended Euclidean algorithm:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\begin{pmatrix}S(x)\Gamma (x)\\x^{6}\end{pmatrix}}\\[6pt]={}&{\begin{pmatrix}\alpha ^{-7}+\alpha ^{4}x+\alpha ^{-1}x^{2}+\alpha ^{6}x^{3}+\alpha ^{-1}x^{4}+\alpha ^{5}x^{5}+\alpha ^{7}x^{6}+\alpha ^{-3}x^{7}\\x^{6}\end{pmatrix}}\\[6pt]={}&{\begin{pmatrix}\alpha ^{7}+\alpha ^{-3}x&1\\1&0\end{pmatrix}}{\begin{pmatrix}x^{6}\\\alpha ^{-7}+\alpha ^{4}x+\alpha ^{-1}x^{2}+\alpha ^{6}x^{3}+\alpha ^{-1}x^{4}+\alpha ^{5}x^{5}+2\alpha ^{7}x^{6}+2\alpha ^{-3}x^{7}\end{pmatrix}}\\[6pt]={}&{\begin{pmatrix}\alpha ^{7}+\alpha ^{-3}x&1\\1&0\end{pmatrix}}{\begin{pmatrix}\alpha ^{4}+\alpha ^{-5}x&1\\1&0\end{pmatrix}}\\&\qquad {\begin{pmatrix}\alpha ^{-7}+\alpha ^{4}x+\alpha ^{-1}x^{2}+\alpha ^{6}x^{3}+\alpha ^{-1}x^{4}+\alpha ^{5}x^{5}\\\alpha ^{-3}+\left(\alpha ^{-7}+\alpha ^{3}\right)x+\left(\alpha ^{3}+\alpha ^{-1}\right)x^{2}+\left(\alpha ^{-5}+\alpha ^{-6}\right)x^{3}+\left(\alpha ^{3}+\alpha ^{1}\right)x^{4}+2\alpha ^{-6}x^{5}+2x^{6}\end{pmatrix}}\\[6pt]={}&{\begin{pmatrix}\left(1+\alpha ^{-4}\right)+\left(\alpha ^{1}+\alpha ^{2}\right)x+\alpha ^{7}x^{2}&\alpha ^{7}+\alpha ^{-3}x\\\alpha ^{4}+\alpha ^{-5}x&1\end{pmatrix}}{\begin{pmatrix}\alpha ^{-7}+\alpha ^{4}x+\alpha ^{-1}x^{2}+\alpha ^{6}x^{3}+\alpha ^{-1}x^{4}+\alpha ^{5}x^{5}\\\alpha ^{-3}+\alpha ^{-2}x+\alpha ^{0}x^{2}+\alpha ^{-2}x^{3}+\alpha ^{-6}x^{4}\end{pmatrix}}\\[6pt]={}&{\begin{pmatrix}\alpha ^{-3}+\alpha ^{5}x+\alpha ^{7}x^{2}&\alpha ^{7}+\alpha ^{-3}x\\\alpha ^{4}+\alpha ^{-5}x&1\end{pmatrix}}{\begin{pmatrix}\alpha ^{-5}+\alpha ^{-4}x&1\\1&0\end{pmatrix}}\\&\qquad {\begin{pmatrix}\alpha ^{-3}+\alpha ^{-2}x+\alpha ^{0}x^{2}+\alpha ^{-2}x^{3}+\alpha ^{-6}x^{4}\\\left(\alpha ^{7}+\alpha ^{-7}\right)+\left(2\alpha ^{-7}+\alpha ^{4}\right)x+\left(\alpha ^{-5}+\alpha ^{-6}+\alpha ^{-1}\right)x^{2}+\left(\alpha ^{-7}+\alpha ^{-4}+\alpha ^{6}\right)x^{3}+\left(\alpha ^{4}+\alpha ^{-6}+\alpha ^{-1}\right)x^{4}+2\alpha ^{5}x^{5}\end{pmatrix}}\\[6pt]={}&{\begin{pmatrix}\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}&\alpha ^{-3}+\alpha ^{5}x+\alpha ^{7}x^{2}\\\alpha ^{3}+\alpha ^{-5}x+\alpha ^{6}x^{2}&\alpha ^{4}+\alpha ^{-5}x\end{pmatrix}}{\begin{pmatrix}\alpha ^{-3}+\alpha ^{-2}x+\alpha ^{0}x^{2}+\alpha ^{-2}x^{3}+\alpha ^{-6}x^{4}\\\alpha ^{-4}+\alpha ^{4}x+\alpha ^{2}x^{2}+\alpha ^{-5}x^{3}\end{pmatrix}}.\end{aligned}}}</annotation>

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</math><img src="./34d562bd03e10536d5a5c4f55c8b831b7dfec00f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -33.171ex; width:121.428ex; height:67.509ex;" alt="{\displaystyle {\begin{aligned}&{\begin{pmatrix}S(x)\Gamma (x)\\x^{6}\end{pmatrix}}\\[6pt]={}&{\begin{pmatrix}\alpha ^{-7}+\alpha ^{4}x+\alpha ^{-1}x^{2}+\alpha ^{6}x^{3}+\alpha ^{-1}x^{4}+\alpha ^{5}x^{5}+\alpha ^{7}x^{6}+\alpha ^{-3}x^{7}\\x^{6}\end{pmatrix}}\\[6pt]={}&{\begin{pmatrix}\alpha ^{7}+\alpha ^{-3}x&1\\1&0\end{pmatrix}}{\begin{pmatrix}x^{6}\\\alpha ^{-7}+\alpha ^{4}x+\alpha ^{-1}x^{2}+\alpha ^{6}x^{3}+\alpha ^{-1}x^{4}+\alpha ^{5}x^{5}+2\alpha ^{7}x^{6}+2\alpha ^{-3}x^{7}\end{pmatrix}}\\[6pt]={}&{\begin{pmatrix}\alpha ^{7}+\alpha ^{-3}x&1\\1&0\end{pmatrix}}{\begin{pmatrix}\alpha ^{4}+\alpha ^{-5}x&1\\1&0\end{pmatrix}}\\&\qquad {\begin{pmatrix}\alpha ^{-7}+\alpha ^{4}x+\alpha ^{-1}x^{2}+\alpha ^{6}x^{3}+\alpha ^{-1}x^{4}+\alpha ^{5}x^{5}\\\alpha ^{-3}+\left(\alpha ^{-7}+\alpha ^{3}\right)x+\left(\alpha ^{3}+\alpha ^{-1}\right)x^{2}+\left(\alpha ^{-5}+\alpha ^{-6}\right)x^{3}+\left(\alpha ^{3}+\alpha ^{1}\right)x^{4}+2\alpha ^{-6}x^{5}+2x^{6}\end{pmatrix}}\\[6pt]={}&{\begin{pmatrix}\left(1+\alpha ^{-4}\right)+\left(\alpha ^{1}+\alpha ^{2}\right)x+\alpha ^{7}x^{2}&\alpha ^{7}+\alpha ^{-3}x\\\alpha ^{4}+\alpha ^{-5}x&1\end{pmatrix}}{\begin{pmatrix}\alpha ^{-7}+\alpha ^{4}x+\alpha ^{-1}x^{2}+\alpha ^{6}x^{3}+\alpha ^{-1}x^{4}+\alpha ^{5}x^{5}\\\alpha ^{-3}+\alpha ^{-2}x+\alpha ^{0}x^{2}+\alpha ^{-2}x^{3}+\alpha ^{-6}x^{4}\end{pmatrix}}\\[6pt]={}&{\begin{pmatrix}\alpha ^{-3}+\alpha ^{5}x+\alpha ^{7}x^{2}&\alpha ^{7}+\alpha ^{-3}x\\\alpha ^{4}+\alpha ^{-5}x&1\end{pmatrix}}{\begin{pmatrix}\alpha ^{-5}+\alpha ^{-4}x&1\\1&0\end{pmatrix}}\\&\qquad {\begin{pmatrix}\alpha ^{-3}+\alpha ^{-2}x+\alpha ^{0}x^{2}+\alpha ^{-2}x^{3}+\alpha ^{-6}x^{4}\\\left(\alpha ^{7}+\alpha ^{-7}\right)+\left(2\alpha ^{-7}+\alpha ^{4}\right)x+\left(\alpha ^{-5}+\alpha ^{-6}+\alpha ^{-1}\right)x^{2}+\left(\alpha ^{-7}+\alpha ^{-4}+\alpha ^{6}\right)x^{3}+\left(\alpha ^{4}+\alpha ^{-6}+\alpha ^{-1}\right)x^{4}+2\alpha ^{5}x^{5}\end{pmatrix}}\\[6pt]={}&{\begin{pmatrix}\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}&\alpha ^{-3}+\alpha ^{5}x+\alpha ^{7}x^{2}\\\alpha ^{3}+\alpha ^{-5}x+\alpha ^{6}x^{2}&\alpha ^{4}+\alpha ^{-5}x\end{pmatrix}}{\begin{pmatrix}\alpha ^{-3}+\alpha ^{-2}x+\alpha ^{0}x^{2}+\alpha ^{-2}x^{3}+\alpha ^{-6}x^{4}\\\alpha ^{-4}+\alpha ^{4}x+\alpha ^{2}x^{2}+\alpha ^{-5}x^{3}\end{pmatrix}}.\end{aligned}}}" loading="lazy"></dd></dl>

We have reached polynomial of degree at most 3, and as

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}-\left(\alpha ^{4}+\alpha ^{-5}x\right)&\alpha ^{-3}+\alpha ^{5}x+\alpha ^{7}x^{2}\\\alpha ^{3}+\alpha ^{-5}x+\alpha ^{6}x^{2}&-\left(\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}\right)\end{pmatrix}}{\begin{pmatrix}\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}&\alpha ^{-3}+\alpha ^{5}x+\alpha ^{7}x^{2}\\\alpha ^{3}+\alpha ^{-5}x+\alpha ^{6}x^{2}&\alpha ^{4}+\alpha ^{-5}x\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}},}</annotation>

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</math><img src="./66b7685a28a1c79648aa9bfb6b1a5665fdd0d40b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:105.546ex; height:7.509ex;" alt="{\displaystyle {\begin{pmatrix}-\left(\alpha ^{4}+\alpha ^{-5}x\right)&\alpha ^{-3}+\alpha ^{5}x+\alpha ^{7}x^{2}\\\alpha ^{3}+\alpha ^{-5}x+\alpha ^{6}x^{2}&-\left(\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}\right)\end{pmatrix}}{\begin{pmatrix}\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}&\alpha ^{-3}+\alpha ^{5}x+\alpha ^{7}x^{2}\\\alpha ^{3}+\alpha ^{-5}x+\alpha ^{6}x^{2}&\alpha ^{4}+\alpha ^{-5}x\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}},}" loading="lazy"></dd></dl>

we get

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}-\left(\alpha ^{4}+\alpha ^{-5}x\right)&\alpha ^{-3}+\alpha ^{5}x+\alpha ^{7}x^{2}\\\alpha ^{3}+\alpha ^{-5}x+\alpha ^{6}x^{2}&-\left(\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}\right)\end{pmatrix}}{\begin{pmatrix}S(x)\Gamma (x)\\x^{6}\end{pmatrix}}={\begin{pmatrix}\alpha ^{-3}+\alpha ^{-2}x+\alpha ^{0}x^{2}+\alpha ^{-2}x^{3}+\alpha ^{-6}x^{4}\\\alpha ^{-4}+\alpha ^{4}x+\alpha ^{2}x^{2}+\alpha ^{-5}x^{3}\end{pmatrix}}.}</annotation>

</semantics>

</math><img src="./37737ee6ce10399a8d705d125fd26e757d2c4361.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:107.77ex; height:7.509ex;" alt="{\displaystyle {\begin{pmatrix}-\left(\alpha ^{4}+\alpha ^{-5}x\right)&\alpha ^{-3}+\alpha ^{5}x+\alpha ^{7}x^{2}\\\alpha ^{3}+\alpha ^{-5}x+\alpha ^{6}x^{2}&-\left(\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}\right)\end{pmatrix}}{\begin{pmatrix}S(x)\Gamma (x)\\x^{6}\end{pmatrix}}={\begin{pmatrix}\alpha ^{-3}+\alpha ^{-2}x+\alpha ^{0}x^{2}+\alpha ^{-2}x^{3}+\alpha ^{-6}x^{4}\\\alpha ^{-4}+\alpha ^{4}x+\alpha ^{2}x^{2}+\alpha ^{-5}x^{3}\end{pmatrix}}.}" loading="lazy"></dd></dl>

Therefore,

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<annotation encoding="application/x-tex">{\displaystyle S(x)\Gamma (x)\left(\alpha ^{3}+\alpha ^{-5}x+\alpha ^{6}x^{2}\right)-\left(\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}\right)x^{6}=\alpha ^{-4}+\alpha ^{4}x+\alpha ^{2}x^{2}+\alpha ^{-5}x^{3}.}</annotation>

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</math><img src="./193a08ebddf760c7fad05fcacde5f5d94dfc82ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:88.279ex; height:3.343ex;" alt="{\displaystyle S(x)\Gamma (x)\left(\alpha ^{3}+\alpha ^{-5}x+\alpha ^{6}x^{2}\right)-\left(\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}\right)x^{6}=\alpha ^{-4}+\alpha ^{4}x+\alpha ^{2}x^{2}+\alpha ^{-5}x^{3}.}" loading="lazy"></dd></dl>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (x)=\alpha ^{3}+\alpha ^{-5}x+\alpha ^{6}x^{2}.}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)=\alpha ^{3}+\alpha ^{-5}x+\alpha ^{6}x^{2}.}</annotation>

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</math><img src="./ad8275e64e5e64f7d94b91f9e8eef9c3f915f762.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.796ex; height:3.176ex;" alt="{\displaystyle \Lambda (x)=\alpha ^{3}+\alpha ^{-5}x+\alpha ^{6}x^{2}.}" loading="lazy"> Don't worry that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{0}\neq 1.}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{0}\neq 1.}</annotation>

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</math><img src="./b2beee877aefd96d5408a54a1819ded4ba481a60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.317ex; height:2.676ex;" alt="{\displaystyle \lambda _{0}\neq 1.}" loading="lazy"> Find by brute force a root of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda .}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda .}</annotation>

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</math><img src="./69f31ae3ac22be0a40c4c1321e4792374285282e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.26ex; height:2.176ex;" alt="{\displaystyle \Lambda .}" loading="lazy"> The roots are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{2},}">

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</math><img src="./e3cec7e4711eca9569982da128a1b5186ae022e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.189ex; height:3.009ex;" alt="{\displaystyle \alpha ^{2},}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{10}}">

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</math><img src="./dc97ca679ffafabbd3d67ec2939f0d71bb80725f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.364ex; height:2.676ex;" alt="{\displaystyle \alpha ^{10}}" loading="lazy"> (after finding for example <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{2}}">

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</math><img src="./9850832395fd0deec6790b4b2265df6656f08ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.542ex; height:2.676ex;" alt="{\displaystyle \alpha ^{2}}" loading="lazy"> we can divide <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

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</math><img src="./0ac0a4a98a414e3480335f9ba652d12571ec6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" loading="lazy"> by corresponding monom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x-\alpha ^{2}\right)}">

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Let

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Xi (x)&=\Gamma (x)\Lambda (x)=\alpha ^{3}+\alpha ^{4}x^{2}+\alpha ^{2}x^{3}+\alpha ^{-5}x^{4}\\\Omega (x)&=S(x)\Xi (x)\equiv \alpha ^{-4}+\alpha ^{4}x+\alpha ^{2}x^{2}+\alpha ^{-5}x^{3}{\bmod {x^{6}}}\end{aligned}}}</annotation>

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</math><img src="./e801a99b618ece595f36831ef5d64e5aa185ba80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:56.501ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\Xi (x)&=\Gamma (x)\Lambda (x)=\alpha ^{3}+\alpha ^{4}x^{2}+\alpha ^{2}x^{3}+\alpha ^{-5}x^{4}\\\Omega (x)&=S(x)\Xi (x)\equiv \alpha ^{-4}+\alpha ^{4}x+\alpha ^{2}x^{2}+\alpha ^{-5}x^{3}{\bmod {x^{6}}}\end{aligned}}}" loading="lazy"></dd></dl>

Let us look for error values using formula

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<annotation encoding="application/x-tex">{\displaystyle e_{j}=-{\frac {\Omega \left(\alpha ^{-i_{j}}\right)}{\Xi '\left(\alpha ^{-i_{j}}\right)}},}</annotation>

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</math><img src="./aafe01967a7a8b69d91e0fdea1532d57ac3d352a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.415ex; height:7.343ex;" alt="{\displaystyle e_{j}=-{\frac {\Omega \left(\alpha ^{-i_{j}}\right)}{\Xi '\left(\alpha ^{-i_{j}}\right)}},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{-i_{j}}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha ^{-i_{j}}}</annotation>

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</math><img src="./b7dda7ff4dd51c9b0ec9e97f81de0562b492bd73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.28ex; height:2.676ex;" alt="{\displaystyle \alpha ^{-i_{j}}}" loading="lazy"> are roots of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Xi (x).}">

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</math><img src="./240a89338100374a3ad0236950105cbac756d177.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.336ex; height:2.843ex;" alt="{\displaystyle \Xi (x).}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Xi '(x)=\alpha ^{2}x^{2}.}">

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<annotation encoding="application/x-tex">{\displaystyle \Xi '(x)=\alpha ^{2}x^{2}.}</annotation>

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</math><img src="./104f96d8f588af3a9360413e7ef5a2769d7777a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.045ex; height:3.176ex;" alt="{\displaystyle \Xi '(x)=\alpha ^{2}x^{2}.}" loading="lazy"> We get

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e_{1}&=-{\frac {\Omega (\alpha ^{4})}{\Xi '(\alpha ^{4})}}={\frac {\alpha ^{-4}+\alpha ^{-7}+\alpha ^{-5}+\alpha ^{7}}{\alpha ^{-5}}}={\frac {\alpha ^{-5}}{\alpha ^{-5}}}=1\\e_{2}&=-{\frac {\Omega (\alpha ^{7})}{\Xi '(\alpha ^{7})}}={\frac {\alpha ^{-4}+\alpha ^{-4}+\alpha ^{1}+\alpha ^{1}}{\alpha ^{1}}}=0\\e_{3}&=-{\frac {\Omega (\alpha ^{10})}{\Xi '(\alpha ^{10})}}={\frac {\alpha ^{-4}+\alpha ^{-1}+\alpha ^{7}+\alpha ^{-5}}{\alpha ^{7}}}={\frac {\alpha ^{7}}{\alpha ^{7}}}=1\\e_{4}&=-{\frac {\Omega (\alpha ^{2})}{\Xi '(\alpha ^{2})}}={\frac {\alpha ^{-4}+\alpha ^{6}+\alpha ^{6}+\alpha ^{1}}{\alpha ^{6}}}={\frac {\alpha ^{6}}{\alpha ^{6}}}=1\end{aligned}}}</annotation>

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</math><img src="./dd99ed2b196802bcd3c91e04ddb2d329e113d47d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:53.693ex; height:27.176ex;" alt="{\displaystyle {\begin{aligned}e_{1}&=-{\frac {\Omega (\alpha ^{4})}{\Xi '(\alpha ^{4})}}={\frac {\alpha ^{-4}+\alpha ^{-7}+\alpha ^{-5}+\alpha ^{7}}{\alpha ^{-5}}}={\frac {\alpha ^{-5}}{\alpha ^{-5}}}=1\\e_{2}&=-{\frac {\Omega (\alpha ^{7})}{\Xi '(\alpha ^{7})}}={\frac {\alpha ^{-4}+\alpha ^{-4}+\alpha ^{1}+\alpha ^{1}}{\alpha ^{1}}}=0\\e_{3}&=-{\frac {\Omega (\alpha ^{10})}{\Xi '(\alpha ^{10})}}={\frac {\alpha ^{-4}+\alpha ^{-1}+\alpha ^{7}+\alpha ^{-5}}{\alpha ^{7}}}={\frac {\alpha ^{7}}{\alpha ^{7}}}=1\\e_{4}&=-{\frac {\Omega (\alpha ^{2})}{\Xi '(\alpha ^{2})}}={\frac {\alpha ^{-4}+\alpha ^{6}+\alpha ^{6}+\alpha ^{1}}{\alpha ^{6}}}={\frac {\alpha ^{6}}{\alpha ^{6}}}=1\end{aligned}}}" loading="lazy"></dd></dl>

Fact, that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{3}=e_{4}=1,}">

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<annotation encoding="application/x-tex">{\displaystyle e_{3}=e_{4}=1,}</annotation>

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</math><img src="./5d5450d9a8cef1fb33b75e977671b37c7adb1ee7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.282ex; height:2.509ex;" alt="{\displaystyle e_{3}=e_{4}=1,}" loading="lazy"> should not be surprising.

Corrected code is therefore [ 1 1 0 1 1 1 0 0 0 0 1 0 1 0 0].

<div class="mw-heading mw-heading4"><h4 id="Decoding_with_unreadable_characters_with_a_small_number_of_errors">Decoding with unreadable characters with a small number of errors</h4></div>

Let us show the algorithm behaviour for the case with small number of errors. Let the received word is [ 1 0 0 ? 1 1 ? 0 0 0 1 0 1 0 0 ].

Again, replace the unreadable characters by zeros while creating the polynomial reflecting their positions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (x)=\left(\alpha ^{8}x-1\right)\left(\alpha ^{11}x-1\right).}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (x)=\left(\alpha ^{8}x-1\right)\left(\alpha ^{11}x-1\right).}</annotation>

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Compute the syndromes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1}=\alpha ^{4},s_{2}=\alpha ^{-7},s_{3}=\alpha ^{1},s_{4}=\alpha ^{1},s_{5}=\alpha ^{0},}">

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<annotation encoding="application/x-tex">{\displaystyle s_{1}=\alpha ^{4},s_{2}=\alpha ^{-7},s_{3}=\alpha ^{1},s_{4}=\alpha ^{1},s_{5}=\alpha ^{0},}</annotation>

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</math><img src="./9fd30fef15b2e272832011744ea9302451a129a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:44.986ex; height:3.009ex;" alt="{\displaystyle s_{1}=\alpha ^{4},s_{2}=\alpha ^{-7},s_{3}=\alpha ^{1},s_{4}=\alpha ^{1},s_{5}=\alpha ^{0},}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{6}=\alpha ^{2}.}">

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<annotation encoding="application/x-tex">{\displaystyle s_{6}=\alpha ^{2}.}</annotation>

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Create syndrome polynomial

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S(x)&=\alpha ^{4}+\alpha ^{-7}x+\alpha ^{1}x^{2}+\alpha ^{1}x^{3}+\alpha ^{0}x^{4}+\alpha ^{2}x^{5},\\S(x)\Gamma (x)&=\alpha ^{4}+\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}+\alpha ^{1}x^{4}+\alpha ^{-1}x^{5}+\alpha ^{-1}x^{6}+\alpha ^{6}x^{7}.\end{aligned}}}</annotation>

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</math><img src="./baa576e72e25421e823194f60a07f0fb38324bd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:72.135ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}S(x)&=\alpha ^{4}+\alpha ^{-7}x+\alpha ^{1}x^{2}+\alpha ^{1}x^{3}+\alpha ^{0}x^{4}+\alpha ^{2}x^{5},\\S(x)\Gamma (x)&=\alpha ^{4}+\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}+\alpha ^{1}x^{4}+\alpha ^{-1}x^{5}+\alpha ^{-1}x^{6}+\alpha ^{6}x^{7}.\end{aligned}}}" loading="lazy"></dd></dl>

Let us run the extended Euclidean algorithm:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\begin{pmatrix}S(x)\Gamma (x)\\x^{6}\end{pmatrix}}&={\begin{pmatrix}\alpha ^{4}+\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}+\alpha ^{1}x^{4}+\alpha ^{-1}x^{5}+\alpha ^{-1}x^{6}+\alpha ^{6}x^{7}\\x^{6}\end{pmatrix}}\\&={\begin{pmatrix}\alpha ^{-1}+\alpha ^{6}x&1\\1&0\end{pmatrix}}{\begin{pmatrix}x^{6}\\\alpha ^{4}+\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}+\alpha ^{1}x^{4}+\alpha ^{-1}x^{5}+2\alpha ^{-1}x^{6}+2\alpha ^{6}x^{7}\end{pmatrix}}\\&={\begin{pmatrix}\alpha ^{-1}+\alpha ^{6}x&1\\1&0\end{pmatrix}}{\begin{pmatrix}\alpha ^{3}+\alpha ^{1}x&1\\1&0\end{pmatrix}}{\begin{pmatrix}\alpha ^{4}+\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}+\alpha ^{1}x^{4}+\alpha ^{-1}x^{5}\\\alpha ^{7}+\left(\alpha ^{-5}+\alpha ^{5}\right)x+2\alpha ^{-7}x^{2}+2\alpha ^{6}x^{3}+2\alpha ^{4}x^{4}+2\alpha ^{2}x^{5}+2x^{6}\end{pmatrix}}\\&={\begin{pmatrix}\left(1+\alpha ^{2}\right)+\left(\alpha ^{0}+\alpha ^{-6}\right)x+\alpha ^{7}x^{2}&\alpha ^{-1}+\alpha ^{6}x\\\alpha ^{3}+\alpha ^{1}x&1\end{pmatrix}}{\begin{pmatrix}\alpha ^{4}+\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}+\alpha ^{1}x^{4}+\alpha ^{-1}x^{5}\\\alpha ^{7}+\alpha ^{0}x\end{pmatrix}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./14d3c176084f36221fedca3e83793155074bccb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.671ex; width:118.338ex; height:26.509ex;" alt="{\displaystyle {\begin{aligned}{\begin{pmatrix}S(x)\Gamma (x)\\x^{6}\end{pmatrix}}&={\begin{pmatrix}\alpha ^{4}+\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}+\alpha ^{1}x^{4}+\alpha ^{-1}x^{5}+\alpha ^{-1}x^{6}+\alpha ^{6}x^{7}\\x^{6}\end{pmatrix}}\\&={\begin{pmatrix}\alpha ^{-1}+\alpha ^{6}x&1\\1&0\end{pmatrix}}{\begin{pmatrix}x^{6}\\\alpha ^{4}+\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}+\alpha ^{1}x^{4}+\alpha ^{-1}x^{5}+2\alpha ^{-1}x^{6}+2\alpha ^{6}x^{7}\end{pmatrix}}\\&={\begin{pmatrix}\alpha ^{-1}+\alpha ^{6}x&1\\1&0\end{pmatrix}}{\begin{pmatrix}\alpha ^{3}+\alpha ^{1}x&1\\1&0\end{pmatrix}}{\begin{pmatrix}\alpha ^{4}+\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}+\alpha ^{1}x^{4}+\alpha ^{-1}x^{5}\\\alpha ^{7}+\left(\alpha ^{-5}+\alpha ^{5}\right)x+2\alpha ^{-7}x^{2}+2\alpha ^{6}x^{3}+2\alpha ^{4}x^{4}+2\alpha ^{2}x^{5}+2x^{6}\end{pmatrix}}\\&={\begin{pmatrix}\left(1+\alpha ^{2}\right)+\left(\alpha ^{0}+\alpha ^{-6}\right)x+\alpha ^{7}x^{2}&\alpha ^{-1}+\alpha ^{6}x\\\alpha ^{3}+\alpha ^{1}x&1\end{pmatrix}}{\begin{pmatrix}\alpha ^{4}+\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}+\alpha ^{1}x^{4}+\alpha ^{-1}x^{5}\\\alpha ^{7}+\alpha ^{0}x\end{pmatrix}}\end{aligned}}}" loading="lazy"></dd></dl>

We have reached polynomial of degree at most 3, and as

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}-1&\alpha ^{-1}+\alpha ^{6}x\\\alpha ^{3}+\alpha ^{1}x&-\left(\alpha ^{-7}+\alpha ^{7}x+\alpha ^{7}x^{2}\right)\end{pmatrix}}{\begin{pmatrix}\alpha ^{-7}+\alpha ^{7}x+\alpha ^{7}x^{2}&\alpha ^{-1}+\alpha ^{6}x\\\alpha ^{3}+\alpha ^{1}x&1\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}},}</annotation>

</semantics>

</math><img src="./f1089f595c1572acef6e79788d7f8f7a17eecd98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:86.264ex; height:6.509ex;" alt="{\displaystyle {\begin{pmatrix}-1&\alpha ^{-1}+\alpha ^{6}x\\\alpha ^{3}+\alpha ^{1}x&-\left(\alpha ^{-7}+\alpha ^{7}x+\alpha ^{7}x^{2}\right)\end{pmatrix}}{\begin{pmatrix}\alpha ^{-7}+\alpha ^{7}x+\alpha ^{7}x^{2}&\alpha ^{-1}+\alpha ^{6}x\\\alpha ^{3}+\alpha ^{1}x&1\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}},}" loading="lazy"></dd></dl>

we get

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}-1&\alpha ^{-1}+\alpha ^{6}x\\\alpha ^{3}+\alpha ^{1}x&-\left(\alpha ^{-7}+\alpha ^{7}x+\alpha ^{7}x^{2}\right)\end{pmatrix}}{\begin{pmatrix}S(x)\Gamma (x)\\x^{6}\end{pmatrix}}={\begin{pmatrix}\alpha ^{4}+\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}+\alpha ^{1}x^{4}+\alpha ^{-1}x^{5}\\\alpha ^{7}+\alpha ^{0}x\end{pmatrix}}.}</annotation>

</semantics>

</math><img src="./5aa1e09febb9ce2d10005c8a4763557606aa8611.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:101.29ex; height:6.509ex;" alt="{\displaystyle {\begin{pmatrix}-1&\alpha ^{-1}+\alpha ^{6}x\\\alpha ^{3}+\alpha ^{1}x&-\left(\alpha ^{-7}+\alpha ^{7}x+\alpha ^{7}x^{2}\right)\end{pmatrix}}{\begin{pmatrix}S(x)\Gamma (x)\\x^{6}\end{pmatrix}}={\begin{pmatrix}\alpha ^{4}+\alpha ^{7}x+\alpha ^{5}x^{2}+\alpha ^{3}x^{3}+\alpha ^{1}x^{4}+\alpha ^{-1}x^{5}\\\alpha ^{7}+\alpha ^{0}x\end{pmatrix}}.}" loading="lazy"></dd></dl>

Therefore,

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<annotation encoding="application/x-tex">{\displaystyle S(x)\Gamma (x)\left(\alpha ^{3}+\alpha ^{1}x\right)-\left(\alpha ^{-7}+\alpha ^{7}x+\alpha ^{7}x^{2}\right)x^{6}=\alpha ^{7}+\alpha ^{0}x.}</annotation>

</semantics>

</math><img src="./c6b24787c67e3bb0666466933c94b78e02535c0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:60.04ex; height:3.343ex;" alt="{\displaystyle S(x)\Gamma (x)\left(\alpha ^{3}+\alpha ^{1}x\right)-\left(\alpha ^{-7}+\alpha ^{7}x+\alpha ^{7}x^{2}\right)x^{6}=\alpha ^{7}+\alpha ^{0}x.}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)=\alpha ^{3}+\alpha ^{1}x.}</annotation>

</semantics>

</math><img src="./244578a63f94cad8f1b38db44d4ec5b101e8d1ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.751ex; height:3.176ex;" alt="{\displaystyle \Lambda (x)=\alpha ^{3}+\alpha ^{1}x.}" loading="lazy"> Don't worry that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{0}\neq 1.}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{0}\neq 1.}</annotation>

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</math><img src="./b2beee877aefd96d5408a54a1819ded4ba481a60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.317ex; height:2.676ex;" alt="{\displaystyle \lambda _{0}\neq 1.}" loading="lazy"> The root of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (x)}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (x)}</annotation>

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</math><img src="./492c958ae36a2bd8fced1b0da90bb6f551fe5bf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.752ex; height:2.843ex;" alt="{\displaystyle \Lambda (x)}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{3-1}.}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha ^{3-1}.}</annotation>

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</math><img src="./b668cb07c79871a4d467c0577810369d778bd6b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.289ex; height:2.676ex;" alt="{\displaystyle \alpha ^{3-1}.}" loading="lazy">

Let

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Xi (x)&=\Gamma (x)\Lambda (x)=\alpha ^{3}+\alpha ^{-7}x+\alpha ^{-4}x^{2}+\alpha ^{5}x^{3},\\\Omega (x)&=S(x)\Xi (x)\equiv \alpha ^{7}+\alpha ^{0}x{\bmod {x^{6}}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./964b2ae260bb9e6794b5db9b767535eceea051a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.099ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\Xi (x)&=\Gamma (x)\Lambda (x)=\alpha ^{3}+\alpha ^{-7}x+\alpha ^{-4}x^{2}+\alpha ^{5}x^{3},\\\Omega (x)&=S(x)\Xi (x)\equiv \alpha ^{7}+\alpha ^{0}x{\bmod {x^{6}}}\end{aligned}}}" loading="lazy"></dd></dl>

Let us look for error values using formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{j}=-\Omega \left(\alpha ^{-i_{j}}\right)/\Xi '\left(\alpha ^{-i_{j}}\right),}">

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<annotation encoding="application/x-tex">{\displaystyle e_{j}=-\Omega \left(\alpha ^{-i_{j}}\right)/\Xi '\left(\alpha ^{-i_{j}}\right),}</annotation>

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</math><img src="./ab1a5827c7c2a033dfde119a344edc441549002a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.99ex; height:3.343ex;" alt="{\displaystyle e_{j}=-\Omega \left(\alpha ^{-i_{j}}\right)/\Xi '\left(\alpha ^{-i_{j}}\right),}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{-i_{j}}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha ^{-i_{j}}}</annotation>

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</math><img src="./b7dda7ff4dd51c9b0ec9e97f81de0562b492bd73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.28ex; height:2.676ex;" alt="{\displaystyle \alpha ^{-i_{j}}}" loading="lazy"> are roots of polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Xi (x).}">

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<annotation encoding="application/x-tex">{\displaystyle \Xi (x).}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \Xi '(x)=\alpha ^{-7}+\alpha ^{5}x^{2}.}</annotation>

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</math><img src="./a5a530b5390b21a43298d3c66a08435f86f1a237.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.706ex; height:3.176ex;" alt="{\displaystyle \Xi '(x)=\alpha ^{-7}+\alpha ^{5}x^{2}.}" loading="lazy"></dd></dl>

We get

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e_{1}&=-{\frac {\Omega \left(\alpha ^{4}\right)}{\Xi '\left(\alpha ^{4}\right)}}={\frac {\alpha ^{7}+\alpha ^{4}}{\alpha ^{-7}+\alpha ^{-2}}}={\frac {\alpha ^{3}}{\alpha ^{3}}}=1\\e_{2}&=-{\frac {\Omega \left(\alpha ^{7}\right)}{\Xi '\left(\alpha ^{7}\right)}}={\frac {\alpha ^{7}+\alpha ^{7}}{\alpha ^{-7}+\alpha ^{4}}}=0\\e_{3}&=-{\frac {\Omega \left(\alpha ^{2}\right)}{\Xi '\left(\alpha ^{2}\right)}}={\frac {\alpha ^{7}+\alpha ^{2}}{\alpha ^{-7}+\alpha ^{-6}}}={\frac {\alpha ^{-3}}{\alpha ^{-3}}}=1\end{aligned}}}</annotation>

</semantics>

</math><img src="./3c2b4dc975e0f4dfca10ee42373e44e98174eae2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.005ex; width:42.037ex; height:21.176ex;" alt="{\displaystyle {\begin{aligned}e_{1}&=-{\frac {\Omega \left(\alpha ^{4}\right)}{\Xi '\left(\alpha ^{4}\right)}}={\frac {\alpha ^{7}+\alpha ^{4}}{\alpha ^{-7}+\alpha ^{-2}}}={\frac {\alpha ^{3}}{\alpha ^{3}}}=1\\e_{2}&=-{\frac {\Omega \left(\alpha ^{7}\right)}{\Xi '\left(\alpha ^{7}\right)}}={\frac {\alpha ^{7}+\alpha ^{7}}{\alpha ^{-7}+\alpha ^{4}}}=0\\e_{3}&=-{\frac {\Omega \left(\alpha ^{2}\right)}{\Xi '\left(\alpha ^{2}\right)}}={\frac {\alpha ^{7}+\alpha ^{2}}{\alpha ^{-7}+\alpha ^{-6}}}={\frac {\alpha ^{-3}}{\alpha ^{-3}}}=1\end{aligned}}}" loading="lazy"></dd></dl>

The fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{3}=1}">

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<annotation encoding="application/x-tex">{\displaystyle e_{3}=1}</annotation>

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</math><img src="./8fc8ced91cf11c0f9fdaf254f278e54945dd52e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.399ex; height:2.509ex;" alt="{\displaystyle e_{3}=1}" loading="lazy"> should not be surprising.

<div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2></div>

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</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;">

<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFReedChen1999">Reed & Chen 1999</a>, p. 189

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFHocquenghem1959">Hocquenghem 1959</a>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFBoseRay-Chaudhuri1960">Bose & Ray-Chaudhuri 1960</a>

</li>

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ipnpr.jpl.nasa.gov/progress_report/42-94/94V.PDF">"Phobos Lander Coding System: Software and Analysis"</a> (PDF). <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://ipnpr.jpl.nasa.gov/progress_report/42-94/94V.PDF">Archived</a> (PDF) from the original on 2022-10-09. Retrieved 25 February 2012.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFMarelliMicheloni2018" class="citation book cs1">Marelli, Alessia; Micheloni, Rino (2018). <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-981-13-0599-3_11">"BCH Codes for Solid-State-Drives"</a>. Inside Solid State Drives (SSDS). Springer Series in Advanced Microelectronics. Vol. 37. pp. 369–406. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-981-13-0599-3_11">10.1007/978-981-13-0599-3_11</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-13-0598-6</bdi>. Retrieved 23 September 2023.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFGilln.d.">Gill n.d.</a>, p. 3

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFLidlPilz1999">Lidl & Pilz 1999</a>, p. 229

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFGorensteinPetersonZierler1960">Gorenstein, Peterson & Zierler 1960</a>

</li>

<li id="cite_note-Gill-Forney-9"><a href="#cite_ref-Gill-Forney_9-0">^</a> <a href="#CITEREFGilln.d.">Gill n.d.</a>, p. 47

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> Yasuo Sugiyama, Masao Kasahara, Shigeichi Hirasawa, and Toshihiko Namekawa. A method for solving key equation for decoding Goppa codes. Information and Control, 27:87–99, 1975.

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Primary_sources">Primary sources</h3></div>

<ul><li><cite id="CITEREFHocquenghem1959" class="citation cs2 cs1-prop-foreign-lang-source"><a href="Alexis_Hocquenghem" title="Alexis Hocquenghem">Hocquenghem, A.</a> (September 1959), "Codes correcteurs d'erreurs", Chiffres (in French), 2, Paris: 147–156</cite></li>

<li><cite id="CITEREFBoseRay-Chaudhuri1960" class="citation cs2"><a href="R._C._Bose" class="mw-redirect" title="R. C. Bose">Bose, R. C.</a>; <a href="D._K._Ray-Chaudhuri" class="mw-redirect" title="D. K. Ray-Chaudhuri">Ray-Chaudhuri, D. K.</a> (March 1960), <a rel="nofollow" class="external text" href="http://repository.lib.ncsu.edu/bitstream/1840.4/2137/1/ISMS_1959_240.pdf">"On A Class of Error Correcting Binary Group Codes"</a> (PDF), <a href="Information_and_Control" class="mw-redirect" title="Information and Control">Information and Control</a>, 3 (1): 68–79, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0019-9958%2860%2990287-4">10.1016/s0019-9958(60)90287-4</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0890-5401">0890-5401</a>, <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://repository.lib.ncsu.edu/bitstream/1840.4/2137/1/ISMS_1959_240.pdf">archived</a> (PDF) from the original on 2022-10-09</cite></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Secondary_sources">Secondary sources</h3></div>

<ul><li><cite id="CITEREFGilln.d." class="citation cs2">Gill, John (n.d.), <a rel="nofollow" class="external text" href="http://www.stanford.edu/class/ee387/handouts/notes7.pdf">EE387 Notes #7, Handout #28</a> (PDF), Stanford University, pp. 42–45, <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www.stanford.edu/class/ee387/handouts/notes7.pdf">archived</a> (PDF) from the original on 2022-10-09, retrieved April 21, 2010</cite> Course notes are apparently being redone for 2012: <a rel="nofollow" class="external free" href="http://www.stanford.edu/class/ee387/">http://www.stanford.edu/class/ee387/</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130605170343/http://www.stanford.edu/class/ee387/">Archived</a> 2013-06-05 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li>

<li><cite id="CITEREFGorensteinPetersonZierler1960" class="citation cs2"><a href="Daniel_Gorenstein" title="Daniel Gorenstein">Gorenstein, Daniel</a>; <a href="W._Wesley_Peterson" title="W. Wesley Peterson">Peterson, W. Wesley</a>; Zierler, Neal (1960), "Two-Error Correcting Bose-Chaudhuri Codes are Quasi-Perfect", Information and Control, 3 (3): 291–294, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0019-9958%2860%2990877-9">10.1016/s0019-9958(60)90877-9</a></cite></li>

<li><cite id="CITEREFLidlPilz1999" class="citation cs2">Lidl, Rudolf; Pilz, Günter (1999), Applied Abstract Algebra (2nd ed.), John Wiley</cite></li>

<li><cite id="CITEREFReedChen1999" class="citation cs2"><a href="Irving_S._Reed" title="Irving S. Reed">Reed, Irving S.</a>; Chen, Xuemin (1999), Error-Control Coding for Data Networks, Boston, MA: <a href="Kluwer_Academic_Publishers" class="mw-redirect" title="Kluwer Academic Publishers">Kluwer Academic Publishers</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7923-8528-4</bdi></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFBlahut2003" class="citation cs2"><a href="Richard_Blahut" title="Richard Blahut">Blahut, Richard E.</a> (2003), Algebraic Codes for Data Transmission (2nd ed.), <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-55374-1</bdi></cite></li>

<li><cite id="CITEREFGilbertNicholson2004" class="citation cs2">Gilbert, W. J.; Nicholson, W. K. (2004), Modern Algebra with Applications (2nd ed.), John Wiley</cite></li>

<li><cite id="CITEREFLinCostello2004" class="citation cs2">Lin, S.; Costello, D. (2004), Error Control Coding: Fundamentals and Applications, Englewood Cliffs, NJ: Prentice-Hall</cite></li>

<li><cite id="CITEREFMacWilliamsSloane1977" class="citation cs2">MacWilliams, F. J.; <a href="N._J._A._Sloane" class="mw-redirect" title="N. J. A. Sloane">Sloane, N. J. A.</a> (1977), The Theory of Error-Correcting Codes, New York, NY: North-Holland Publishing Company</cite></li>

<li><cite id="CITEREFRudra" class="citation cs2">Rudra, Atri, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121218004156/http://www.cse.buffalo.edu:80/~atri/courses/coding-theory/">CSE 545, Error Correcting Codes: Combinatorics, Algorithms and Applications</a>, University at Buffalo, archived from <a rel="nofollow" class="external text" href="http://www.cse.buffalo.edu/~atri/courses/coding-theory/">the original</a> on 2012-12-18</cite></li></ul></div><div><div class="zim-footer">

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