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# Alternant code

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

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== References == | == References == | ||

* {{cite book | author=F.J. MacWilliams | authorlink=Jessie MacWilliams | coauthors=N.J.A. Sloane | title=The Theory of Error-Correcting Codes | publisher=North-Holland | date=1977 | isbn=0-444-85193-3 | pages=332-338 }} | * {{cite book | author=F.J. MacWilliams | authorlink=Jessie MacWilliams | coauthors=N.J.A. Sloane | title=The Theory of Error-Correcting Codes | publisher=North-Holland | date=1977 | isbn=0-444-85193-3 | pages=332-338 }} | ||

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## Latest revision as of 20:15, 29 October 2008

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 *H*_{i,j} = α_{j}^{i}*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-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

- F.J. MacWilliams; N.J.A. Sloane (1977).
*The Theory of Error-Correcting Codes*. North-Holland, 332-338. ISBN 0-444-85193-3.