alternant code

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

{{one source |date=April 2024}}

In coding theory, alternant codes form a class of parameterised error-correcting codes which generalise the BCH codes.

Definition

An alternant code over GF(q) of length n is defined by a parity check matrix H of alternant form Hi,j = αjiyi, where the αj are distinct elements of the extension GF(qm), the yi are further non-zero parameters again in the extension GF(qm) and the indices range as i from 0 to δ − 1, j from 1 to n.

Properties

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

There exist long alternant codes which meet the Gilbert–Varshamov bound.

The class of alternant codes includes

References

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  • {{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] }}

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Category:Error detection and correction

Category:Finite fields

Category:Coding theory

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