Alternant code: Difference between revisions

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{{Short description|Class of error correction code}}

{{one source |date=April 2024}}

In [[coding theory]], '''alternant codes''' form a class of parameterised [[Error detection and correction|error-correcting codes]] which generalise the [[BCH code]]s.

==Definition==

An ''alternant code'' over GF(''q'') of length ''n'' is defined by a parity check matrix ''H'' of [[alternant matrix|alternant]] form ''H''_''i'',''j'' = α_jⁱ''y''_''i'', where the α_''j'' are distinct elements of the extension GF(''q''^''m''), the ''y''_''i'' are further non-zero parameters again in the extension GF(''q''^''m'') and the indices range as ''i'' from 0 to δ &minunsminus; 1, ''j'' from 1 to ''n''.

==Properties==

The parameters of this alternant code are length ''n'', dimension ≥ ''n-mδ'' − ''m''δ and minimum distance ≥  δ + 1.

There exist long alternant codes which meet the [[Gilbert-VarshamovGilbert–Varshamov bound]].

The class of alternant codes includes

* [[BCH code]]s

* [[Binary Goppa code|Goppa codes]]s

* [[SrivastaSrivastava code]]s

== References ==

{{refbegin}}

* {{cite book | author=F.J. MacWilliams | authorlink=Jessie MacWilliams | coauthorsauthor2=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/332-338 332–338] }}

{{refend}}

[[Category:Error detection and correction]]

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[[Category:Coding theory]]

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