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<title>Systematic code</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Systematic code</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Coding_theory" title="Coding theory">coding theory</a>, a <b>systematic code</b> is any <a href="Error-correcting_code" class="mw-redirect" title="Error-correcting code">error-correcting code</a> in which the input data are embedded in the encoded output. Conversely, in a <b>non-systematic code</b> the output does not contain the input symbols.

</p><p>Systematic codes have the advantage that the parity data can simply be appended to the source block, and receivers do not need to recover the original source symbols if received correctly – this is useful for example if error-correction coding is combined with a hash function for quickly determining the correctness of the received source symbols, or in cases where errors occur in <a href="Erasure_channel" class="mw-redirect" title="Erasure channel">erasures</a> and a received symbol is thus always correct. Furthermore, for engineering purposes such as synchronization and monitoring, it is desirable to get reasonable good estimates of the received source symbols without going through the lengthy decoding process which may be carried out at a remote site at a later time.<sup id="cite_ref-nonsystematic_1-0" class="reference"><a href="#cite_note-nonsystematic-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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

<p>Every non-systematic linear code can be transformed into a systematic code with essentially the same properties (i.e., minimum distance).<sup id="cite_ref-nonsystematic_1-1" class="reference"><a href="#cite_note-nonsystematic-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Richard_Blahut_2-0" class="reference"><a href="#cite_note-Richard_Blahut-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

Because of the advantages cited above, <a href="Linear_code" title="Linear code">linear</a> error-correcting codes are therefore generally implemented as systematic codes. However, for certain decoding algorithms such as sequential decoding or maximum-likelihood decoding, a non-systematic structure can increase performance in terms of undetected decoding error probability when the minimum <i>free</i> distance of the code is larger.<sup id="cite_ref-nonsystematic_1-2" class="reference"><a href="#cite_note-nonsystematic-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>For a systematic <a href="Linear_code" title="Linear code">linear code</a>, the <a href="Generator_matrix" title="Generator matrix">generator matrix</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span>, can always be written as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=[I_{k}|P]}">

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<annotation encoding="application/x-tex">{\displaystyle G=[I_{k}|P]}</annotation>

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</math></span><img src="./3515ee9aa6233a0444be04f483c70d7590d05e6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.723ex; height:2.843ex;" alt="{\displaystyle G=[I_{k}|P]}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{k}}">

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

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</math></span><img src="./d658e7f6b34dd1d3025a7c9a72efba5b9f46475d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.112ex; height:2.509ex;" alt="{\displaystyle I_{k}}" loading="lazy"></span> is the <a href="Identity_matrix" title="Identity matrix">identity matrix</a> of size <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span>.

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

<ul><li><a href="Checksum" title="Checksum">Checksums</a> and <a href="Hash_function" title="Hash function">hash functions</a>, combined with the input data, can be viewed as systematic error-detecting codes.</li>

<li>Linear codes are usually implemented as systematic error-correcting codes (e.g., Reed-Solomon codes in <a href="Compact_Disc" class="mw-redirect" title="Compact Disc">CDs</a>).</li>

<li><a href="Convolutional_code" title="Convolutional code">Convolutional codes</a> are implemented as either systematic or non-systematic codes. Non-systematic convolutional codes can provide better performance under maximum-likelihood (<a href="Viterbi_decoder" title="Viterbi decoder">Viterbi</a>) decoding.</li>

<li>In <a href="DVB-H" title="DVB-H">DVB-H</a>, for additional error protection and power efficiency for mobile receivers, a systematic <a href="Reed%E2%80%93Solomon_error_correction" title="Reed–Solomon error correction">Reed-Solomon code</a> is employed as an erasure code over packets within a <a href="Burst_transmission" title="Burst transmission">data burst</a>, where each packet is protected with a <a href="Cyclic_redundancy_check" title="Cyclic redundancy check">CRC</a>: data in verified packets count as correctly received symbols, and if all are received correctly, evaluation of the additional parity data can be omitted, and receiver devices can switch off reception until the start of the next burst.</li>

<li><a href="Fountain_code" title="Fountain code">Fountain codes</a> may be either systematic or non-systematic: as they do not exhibit a fixed <a href="Code_rate" title="Code rate">code rate</a>, the set of source symbols is diminishing among the possible output set.</li></ul>

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

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</style><cite id="CITEREFJames_L._Massey,_Daniel_J._Costello,_Jr.1971" class="citation journal cs1"><a href="James_L._Massey" class="mw-redirect" title="James L. Massey">James L. Massey</a>, Daniel J. Costello, Jr. (1971). "Nonsystematic convolutional codes for sequential decoding in space applications". <i>IEEE Transactions on Communication Technology</i>. <b>19</b> (5): <span class="nowrap">806–</span>813. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTCOM.1971.1090720">10.1109/TCOM.1971.1090720</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:51650729">51650729</a>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite journal}}</code>: CS1 maint: multiple names: authors list (link)</span></span>

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<li id="cite_note-Richard_Blahut-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Richard_Blahut_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFRichard_E._Blahut2003" class="citation book cs1">Richard E. Blahut (2003). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/algebraiccodesfo00blah"><i>Algebraic codes for data transmission</i></a></span> (2nd ed.). Cambridge. Univ. Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/algebraiccodesfo00blah/page/n68">53</a>–54. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-55374-2</bdi>.</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFShu_LinDaniel_J._Costello,_Jr.1983" class="citation book cs1">Shu Lin; Daniel J. Costello, Jr. (1983). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/errorcontrolcodi00lins_929"><i>Error Control Coding: Fundamentals and Applications</i></a></span>. <a href="Prentice_Hall" title="Prentice Hall">Prentice Hall</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/errorcontrolcodi00lins_929/page/n296">278</a>–280. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-283796-X</bdi>.</cite></span>

</li>

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

<ul><li><cite id="CITEREFShu_LinDaniel_J._Costello,_Jr.1983" class="citation book cs1">Shu Lin; Daniel J. Costello, Jr. (1983). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/errorcontrolcodi00lins_929"><i>Error Control Coding: Fundamentals and Applications</i></a></span>. <a href="Prentice_Hall" title="Prentice Hall">Prentice Hall</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/errorcontrolcodi00lins_929/page/n296">278</a>–280. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-283796-X</bdi>.</cite></li></ul></div><div><div class="zim-footer">

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