<!DOCTYPE html>
<html class="client-nojs vector-feature-night-mode-disabled vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-sticky-header-enabled" lang="en" dir="ltr"><head>
<meta charset="UTF-8">
<title>Code rate</title>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="canonical" href="https://en.wikipedia.org/wiki/Code_rate"> <link href="./mw/ext.cite.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/ext.math.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.icons.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.search.codex.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/user.styles.css" rel="stylesheet" type="text/css">
<meta name="ResourceLoaderDynamicStyles" content="">
<link rel="stylesheet" type="text/css" href="./mw/site.styles.css">
<link rel="stylesheet" type="text/css" href="./mw/noscript.css">
<link rel="stylesheet" type="text/css" href="./footer.css">
<link rel="stylesheet" type="text/css" href="./vector-2022.css">
</head>
<body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Code_rate rootpage-Code_rate skin-vector-2022 action-view">
<div class="mw-page-container">
<div class="mw-page-container-inner">
<div class="mw-content-container">
<main id="content" class="mw-body">
<header class="mw-body-header vector-page-titlebar">
<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Code rate</span></span>
</h1>
</header>
<a id="top"></a>
<div id="bodyContent" class="vector-body ve-init-mw-desktopArticleTarget-targetContainer" aria-labelledby="firstHeading" data-mw-ve-target-container="">
<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">
<style data-mw-deduplicate="TemplateStyles:r1251242444">
/* start https://en.wikipedia.org/ */
.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}
/* end https://en.wikipedia.org/ */
</style>
<p>In <a href="Telecommunication" class="mw-redirect" title="Telecommunication">telecommunication</a> and <a href="Information_theory" title="Information theory">information theory</a>, the <b>code rate</b> (or <b>information rate</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>) of a <a href="Forward_error_correction" class="mw-redirect" title="Forward error correction">forward error correction</a> code is the proportion of the data-stream that is useful (non-<a href="Redundancy_(information_theory)" title="Redundancy (information theory)">redundant</a>). That is, if the code rate is <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/n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k/n}</annotation>
</semantics>
</math></span><img src="./df39d254fa80db61cf468fed3a8c4dbb10e7e7d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.768ex; height:2.843ex;" alt="{\displaystyle k/n}" loading="lazy"></span> for every <span class="texhtml mvar" style="font-style:italic;">k</span> bits of useful information, the coder generates a total of <span class="texhtml mvar" style="font-style:italic;">n</span> bits of data, of which <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-k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n-k}</annotation>
</semantics>
</math></span><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"></span> are redundant.
</p><p>If <span class="texhtml mvar" style="font-style:italic;">R</span> is the <a href="Gross_bit_rate" class="mw-redirect" title="Gross bit rate">gross bit rate</a> or <a href="Data_signalling_rate" class="mw-redirect" title="Data signalling rate">data signalling rate</a> (inclusive of redundant error coding), the <a href="Net_bit_rate" class="mw-redirect" title="Net bit rate">net bit rate</a> (the useful bit rate exclusive of <a href="Error_correction_code" title="Error correction code">error correction codes</a>) is <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 \leq R\cdot k/n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>≤<!-- ≤ --></mo>
<mi>R</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \leq R\cdot k/n}</annotation>
</semantics>
</math></span><img src="./925ed3a53dbdd04ab313b98bd49194e195353f1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.665ex; height:2.843ex;" alt="{\displaystyle \leq R\cdot k/n}" loading="lazy"></span>.
</p><p>For example: The code rate of a <a href="Convolutional_code" title="Convolutional code">convolutional code</a> will typically be <style data-mw-deduplicate="TemplateStyles:r1154941027">
/* start https://en.wikipedia.org/ */
.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}
/* end https://en.wikipedia.org/ */
</style><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>, <span class="frac"><span class="num">2</span>⁄<span class="den">3</span></span>, <span class="frac"><span class="num">3</span>⁄<span class="den">4</span></span>, <span class="frac"><span class="num">5</span>⁄<span class="den">6</span></span>, <span class="frac"><span class="num">7</span>⁄<span class="den">8</span></span>, etc., corresponding to one redundant bit inserted after every single, second, third, etc., bit. The code rate of the <a href="Octet_(computing)" title="Octet (computing)">octet</a> oriented <a href="Reed_Solomon" class="mw-redirect" title="Reed Solomon">Reed Solomon</a> <a href="Block_code" title="Block code">block code</a> denoted RS(204,188) is 188/204, meaning that <span class="texhtml">204 − 188 = 16</span> redundant octets (or bytes) are added to each block of 188 octets of useful information.
</p><p>A few error correction codes do not have a fixed code rate—<a href="Rateless_erasure_code" class="mw-redirect" title="Rateless erasure code">rateless erasure codes</a>.
</p><p>Note that <a href="Bit/s" class="mw-redirect" title="Bit/s">bit/s</a> is a more widespread <a href="Unit_of_measurement" title="Unit of measurement">unit of measurement</a> for the <a href="Information_rate" class="mw-redirect" title="Information rate">information rate</a>, implying that it is synonymous with <i>net bit rate</i> or <i>useful bit rate</i> exclusive of error-correction codes.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Entropy_rate" title="Entropy rate">Entropy rate</a></li>
<li><a href="Information_rate" class="mw-redirect" title="Information rate">Information rate</a></li>
<li><a href="Punctured_code" title="Punctured code">Punctured code</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1239543626">
/* start https://en.wikipedia.org/ */
.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}
/* end https://en.wikipedia.org/ */
</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Huffman, W. Cary, and Pless, Vera, <i>Fundamentals of Error-Correcting Codes</i>, Cambridge, 2003.</span>
</li>
</ol></div></div>
<p><br>
</p>
<style data-mw-deduplicate="TemplateStyles:r1271159938">
/* start https://en.wikipedia.org/ */
.mw-parser-output .asbox{position:relative;overflow:hidden}.mw-parser-output .asbox table{background:transparent}.mw-parser-output .asbox p{margin:0}.mw-parser-output .asbox p+p{margin-top:0.25em}.mw-parser-output .asbox-body{font-style:italic}.mw-parser-output .asbox-note{font-size:smaller}.mw-parser-output .asbox .navbar{position:absolute;top:-0.75em;right:1em;display:none}.mw-parser-output :not(p):not(.asbox)+style+.asbox,.mw-parser-output :not(p):not(.asbox)+link+.asbox{margin-top:3em}
/* end https://en.wikipedia.org/ */
</style></div><!--htdig_noindex--><div><div class="zim-footer">
This article is issued from <a class="external text" title="Last edited on 2024-04-12" href="https://en.wikipedia.org/wiki/?title=Code_rate&oldid=1218520213">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.
</div>
</div><!--/htdig_noindex--></div>
</div>
</main>
</div>
</div>
</div>
</body></html>