http://debianws.lexgopc.com/wiki143/index.php?action=history&feed=atom&title=Alternant_code 2026-05-06T16:26:49Z Revision history for this page on the wiki MediaWiki 1.43.1 http://debianws.lexgopc.com/wiki143/index.php?title=Alternant_code&diff=7066005&oldid=prev 2024-04-28T03:29:16Z

<p>tag as one source</p> <p><b>New page</b></p><div>{{Short description|Class of error correction code}}<br /> {{one source |date=April 2024}}<br /> In [[coding theory]], &#039;&#039;&#039;alternant codes&#039;&#039;&#039; form a class of parameterised [[Error detection and correction|error-correcting codes]] which generalise the [[BCH code]]s.<br /> <br /> ==Definition==<br /> An &#039;&#039;alternant code&#039;&#039; over GF(&#039;&#039;q&#039;&#039;) of length &#039;&#039;n&#039;&#039; is defined by a parity check matrix &#039;&#039;H&#039;&#039; of [[alternant matrix|alternant]] form &#039;&#039;H&#039;&#039;&lt;sub&gt;&#039;&#039;i&#039;&#039;,&#039;&#039;j&#039;&#039;&lt;/sub&gt; = α&lt;sub&gt;j&lt;/sub&gt;&lt;sup&gt;i&lt;/sup&gt;&#039;&#039;y&#039;&#039;&lt;sub&gt;&#039;&#039;i&#039;&#039;&lt;/sub&gt;, where the α&lt;sub&gt;&#039;&#039;j&#039;&#039;&lt;/sub&gt; are distinct elements of the extension GF(&#039;&#039;q&#039;&#039;&lt;sup&gt;&#039;&#039;m&#039;&#039;&lt;/sup&gt;), the &#039;&#039;y&#039;&#039;&lt;sub&gt;&#039;&#039;i&#039;&#039;&lt;/sub&gt; are further non-zero parameters again in the extension GF(&#039;&#039;q&#039;&#039;&lt;sup&gt;&#039;&#039;m&#039;&#039;&lt;/sup&gt;) and the indices range as &#039;&#039;i&#039;&#039; from 0 to δ&amp;nbsp;&amp;minus;&amp;nbsp;1, &#039;&#039;j&#039;&#039; from 1 to &#039;&#039;n&#039;&#039;.<br /> <br /> ==Properties==<br /> The parameters of this alternant code are length &#039;&#039;n&#039;&#039;, dimension ≥ &#039;&#039;n&#039;&#039;&amp;nbsp;&amp;minus;&amp;nbsp;&#039;&#039;m&#039;&#039;δ and minimum distance ≥&amp;nbsp;δ&amp;nbsp;+&amp;nbsp;1.<br /> There exist long alternant codes which meet the [[Gilbert–Varshamov bound]].<br /> <br /> The class of alternant codes includes <br /> * [[BCH code]]s<br /> * [[Binary Goppa code|Goppa codes]]<br /> * [[Srivastava code]]s<br /> <br /> == References ==<br /> {{refbegin}}<br /> * {{cite book | author=F.J. MacWilliams | authorlink=Jessie MacWilliams |author2=N.J.A. Sloane | author2link=Neil Sloane | title=The Theory of Error-Correcting Codes | url=https://archive.org/details/theoryoferrorcor0000macw | url-access=registration | publisher=North-Holland | date=1977 | isbn=0-444-85193-3 | pages=[https://archive.org/details/theoryoferrorcor0000macw/page/332 332–338] }}<br /> {{refend}}<br /> <br /> [[Category:Error detection and correction]]<br /> [[Category:Finite fields]]<br /> [[Category:Coding theory]]<br /> <br /> <br /> {{crypto-stub}}</div>

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