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<title>Sequential decoding</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Sequential decoding</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>Recognised by <a href="John_Wozencraft" title="John Wozencraft">John Wozencraft</a>, <b>sequential decoding</b> is a limited memory technique for decoding tree codes. Sequential decoding is mainly used as an approximate decoding algorithm for long constraint-length <a href="Convolutional_code" title="Convolutional code">convolutional codes</a>. This approach may not be as accurate as the <a href="Viterbi_algorithm" title="Viterbi algorithm">Viterbi algorithm</a> but can save a substantial amount of computer memory. It was used to decode a convolutional code in 1968 <a href="Pioneer_9" class="mw-redirect" title="Pioneer 9">Pioneer 9</a> mission.

</p><p>Sequential decoding explores the tree code in such a way to try to minimise the computational cost and memory requirements to store the tree.

</p><p>There is a range of sequential decoding approaches based on the choice of metric and algorithm. Metrics include:

</p>

<ul><li>Fano metric</li>

<li>Zigangirov metric</li>

<li>Gallager metric</li></ul>

<p>Algorithms include:

</p>

<ul><li>Stack algorithm</li>

<li>Fano algorithm</li>

<li>Creeper algorithm</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Fano_metric">Fano metric</h2></div>

<p>Given a partially explored tree (represented by a set of nodes which are limit of exploration), we would like to know the best node from which to explore further. The Fano metric (named after <a href="Robert_Fano" title="Robert Fano">Robert Fano</a>) allows one to calculate from which is the best node to explore further. This metric is optimal given no other constraints (e.g. memory).

</p><p>For a <a href="Binary_symmetric_channel" title="Binary symmetric channel">binary symmetric channel</a> (with error probability <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 p}">

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

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</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>) the Fano metric can be derived via <a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a>. We are interested in following the most likely path <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 P_{i}}">

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

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</math></span><img src="./3ba1396129f7be3c7f828a571b6649e6807d10d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}" loading="lazy"></span> given an explored state of the tree <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 X}">

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

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> and a received sequence <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 {\mathbf {r} }}">

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

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</math></span><img src="./fb19eb313ea5fd985454fb71fe17b8479f780c45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle {\mathbf {r} }}" loading="lazy"></span>. Using the language of <a href="Probability" title="Probability">probability</a> and <a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a> we want to choose the maximum over <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}">

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

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</math></span><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"></span> of:

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<annotation encoding="application/x-tex">{\displaystyle \Pr(P_{i}|X,{\mathbf {r} })\propto \Pr({\mathbf {r} }|P_{i},X)\Pr(P_{i}|X)}</annotation>

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</math></span><img src="./95f2aa7d171169e152782846acbfade4935446ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.425ex; height:2.843ex;" alt="{\displaystyle \Pr(P_{i}|X,{\mathbf {r} })\propto \Pr({\mathbf {r} }|P_{i},X)\Pr(P_{i}|X)}" loading="lazy"></span></dd></dl>

<p>We now introduce the following notation:

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

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> to represent the maximum length of transmission in branches</li>

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

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</math></span><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span> to represent the number of bits on a branch of the code (the denominator of the <a href="Code_rate" title="Code rate">code rate</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 R}">

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

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</math></span><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"></span>).</li>

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

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</math></span><img src="./abe3154db7d4f92fb42dd1f80f52f528c6312e4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.009ex; height:2.509ex;" alt="{\displaystyle d_{i}}" loading="lazy"></span> to represent the number of bit errors on path <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 P_{i}}">

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

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</math></span><img src="./3ba1396129f7be3c7f828a571b6649e6807d10d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}" loading="lazy"></span> (the <a href="Hamming_distance" title="Hamming distance">Hamming distance</a> between the branch labels and the received sequence)</li>

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

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</math></span><img src="./57f87f905ba5a4d8c691ccaecd65fc47bd007ba4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}" loading="lazy"></span> to be the length of <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 P_{i}}">

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

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</math></span><img src="./3ba1396129f7be3c7f828a571b6649e6807d10d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}" loading="lazy"></span> in branches.</li></ul>

<p>We express the <a href="Likelihood" class="mw-redirect" title="Likelihood">likelihood</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 \Pr({\mathbf {r} }|P_{i},X)}">

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<annotation encoding="application/x-tex">{\displaystyle \Pr({\mathbf {r} }|P_{i},X)}</annotation>

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</math></span><img src="./ffc88d5159747d8518bd71e16d08d2124761787e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.359ex; height:2.843ex;" alt="{\displaystyle \Pr({\mathbf {r} }|P_{i},X)}" loading="lazy"></span> 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 p^{d_{i}}(1-p)^{n_{i}b-d_{i}}2^{-(N-n_{i})b}}">

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<annotation encoding="application/x-tex">{\displaystyle p^{d_{i}}(1-p)^{n_{i}b-d_{i}}2^{-(N-n_{i})b}}</annotation>

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</math></span><img src="./82972429a381b50a59168b2b77497f113db878c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:24.266ex; height:3.343ex;" alt="{\displaystyle p^{d_{i}}(1-p)^{n_{i}b-d_{i}}2^{-(N-n_{i})b}}" loading="lazy"></span> (by using the binary symmetric channel likelihood for the first <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 n_{i}b}">

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

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</math></span><img src="./d2d8d56adbb1cee89c3478e0328e8c286de6569c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.192ex; height:2.509ex;" alt="{\displaystyle n_{i}b}" loading="lazy"></span> bits followed by a uniform prior over the remaining bits).

</p><p>We express the <a href="Prior_probability" title="Prior probability">prior</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 \Pr(P_{i}|X)}">

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<annotation encoding="application/x-tex">{\displaystyle \Pr(P_{i}|X)}</annotation>

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</math></span><img src="./0b379740896f6137ea2173de41f4d4213dc2e9df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.223ex; height:2.843ex;" alt="{\displaystyle \Pr(P_{i}|X)}" loading="lazy"></span> in terms of the number of branch choices one has made, <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 n_{i}}">

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

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</math></span><img src="./57f87f905ba5a4d8c691ccaecd65fc47bd007ba4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}" loading="lazy"></span>, and the number of branches from each node, <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 2^{Rb}}">

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

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</math></span><img src="./48bb0ccdf0b881c165be7953e62fccf9218daca7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.347ex; height:2.676ex;" alt="{\displaystyle 2^{Rb}}" loading="lazy"></span>.

</p><p>Therefore:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Pr(P_{i}|X,{\mathbf {r} })&\propto p^{d_{i}}(1-p)^{n_{i}b-d_{i}}2^{-(N-n_{i})b}2^{-n_{i}Rb}\\&\propto p^{d_{i}}(1-p)^{n_{i}b-d_{i}}2^{n_{i}b}2^{-n_{i}Rb}\end{aligned}}}</annotation>

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</math></span><img src="./af111eebb17b93bb6c84e490d7b9c8346d4a7006.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.622ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\Pr(P_{i}|X,{\mathbf {r} })&\propto p^{d_{i}}(1-p)^{n_{i}b-d_{i}}2^{-(N-n_{i})b}2^{-n_{i}Rb}\\&\propto p^{d_{i}}(1-p)^{n_{i}b-d_{i}}2^{n_{i}b}2^{-n_{i}Rb}\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>We can equivalently maximise the log of this probability, i.e.

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&d_{i}\log _{2}p+(n_{i}b-d_{i})\log _{2}(1-p)+n_{i}b-n_{i}Rb\\=&d_{i}(\log _{2}p+1-R)+(n_{i}b-d_{i})(\log _{2}(1-p)+1-R)\end{aligned}}}</annotation>

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</math></span><img src="./009c4a536d983ac2200846bb1b72f1e0e5756f0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:54.682ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&d_{i}\log _{2}p+(n_{i}b-d_{i})\log _{2}(1-p)+n_{i}b-n_{i}Rb\\=&d_{i}(\log _{2}p+1-R)+(n_{i}b-d_{i})(\log _{2}(1-p)+1-R)\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>This last expression is the Fano metric. The important point to see is that we have two terms here: one based on the number of wrong bits and one based on the number of right bits. We can therefore update the Fano metric simply by adding <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 \log _{2}p+1-R}">

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<annotation encoding="application/x-tex">{\displaystyle \log _{2}p+1-R}</annotation>

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</math></span><img src="./c740d262e784951f2093c04b0b34a7bf4f38b6d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.19ex; height:2.676ex;" alt="{\displaystyle \log _{2}p+1-R}" loading="lazy"></span> for each non-matching bit and <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 \log _{2}(1-p)+1-R}">

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<annotation encoding="application/x-tex">{\displaystyle \log _{2}(1-p)+1-R}</annotation>

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</math></span><img src="./0254f4610379bf22da1aa8da2821f87a064b73ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.615ex; height:2.843ex;" alt="{\displaystyle \log _{2}(1-p)+1-R}" loading="lazy"></span> for each matching bit.

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<div class="mw-heading mw-heading2"><h2 id="Computational_cutoff_rate">Computational cutoff rate</h2></div>

<p>For sequential decoding to be a good choice of decoding algorithm, the number of states explored should remain small (otherwise an algorithm which deliberately explores all states, e.g. the <a href="Viterbi_algorithm" title="Viterbi algorithm">Viterbi algorithm</a>, may be more suitable). For a particular noise level there is a maximum coding rate <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 R_{0}}">

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

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</math></span><img src="./9b8916196f182fcbaaca54f931176a4a4f5769cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{0}}" loading="lazy"></span> called the computational cutoff rate where there is a finite backtracking limit. For the binary symmetric channel:

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<annotation encoding="application/x-tex">{\displaystyle R_{0}=1-\log _{2}(1+2{\sqrt {p(1-p)}})}</annotation>

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</math></span><img src="./68b2ec8d077220cb89ec5a28ad8f849edc60a369.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.395ex; height:4.843ex;" alt="{\displaystyle R_{0}=1-\log _{2}(1+2{\sqrt {p(1-p)}})}" loading="lazy"></span></dd></dl>

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

<div class="mw-heading mw-heading3"><h3 id="Stack_algorithm">Stack algorithm</h3></div>

<p>The simplest algorithm to describe is the "stack algorithm" in which the best <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 N}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> paths found so far are stored. Sequential decoding may introduce an additional error above Viterbi decoding when the correct path has <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 N}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> or more highly scoring paths above it; at this point the best path will drop off the stack and be no longer considered.

</p>

<div class="mw-heading mw-heading3"><h3 id="Fano_algorithm">Fano algorithm</h3></div>

<p>The famous Fano algorithm (named after <a href="Robert_Fano" title="Robert Fano">Robert Fano</a>) has a very low memory requirement and hence is suited to hardware implementations. This algorithm explores backwards and forward from a single point on the tree.

</p>

<ol><li>The Fano algorithm is a sequential decoding algorithm that does not require a stack.</li>

<li>The Fano algorithm can only operate over a code tree because it cannot examine path merging.</li>

<li>At each decoding stage, the Fano algorithm retains the information regarding three paths: the current path, its immediate predecessor path, and one of its successor paths.</li>

<li>Based on this information, the Fano algorithm can move from the current path to either its immediate predecessor path or the selected successor path; hence, no stack is required for queuing all examined paths.</li>

<li>The movement of the Fano algorithm is guided by a dynamic threshold <var style="padding-right: 1px;">T</var> that is an integer multiple of a fixed step size Δ.</li>

<li>Only the path whose path metric is no less than <var style="padding-right: 1px;">T</var> can be next visited. According to the algorithm, the process of codeword search continues to move forward along a code path, as long as the Fano metric along the code path remains non-decreasing.</li>

<li>Once all the successor path metrics are smaller than <var style="padding-right: 1px;">T</var>, the algorithm moves backward to the predecessor path if the predecessor path metric beats <var style="padding-right: 1px;">T</var>; thereafter, threshold examination will be subsequently performed on another successor path of this revisited predecessor.</li>

<li>In case the predecessor path metric is also less than <var style="padding-right: 1px;">T</var>, the threshold <var style="padding-right: 1px;">T</var> is one-step lowered so that the algorithm is not trapped on the current path.</li>

<li>For the Fano algorithm, if a path is revisited, the presently examined dynamic threshold is always lower than the momentary dynamic threshold at the previous visit, guaranteeing that looping in the algorithm does not occur, and that the algorithm can ultimately reach a terminal node of the code tree, and stop.</li></ol>

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

<ul><li><a href="John_Wozencraft" title="John Wozencraft">John Wozencraft</a> and B. Reiffen, <i>Sequential decoding</i>, <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-262-23006-2</bdi></li>

<li>Rolf Johannesson and Kamil Sh. Zigangirov, <i>Fundamentals of convolutional coding</i> (chapter 6), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-470-27683-5</bdi></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li>"<a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/CorrectionTrees/">Correction trees</a>" - simulator of correction process using priority queue to choose maximum metric node (called weight)</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-10" href="https://en.wikipedia.org/wiki/?title=Sequential_decoding&oldid=1284888014">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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