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Alternant code
From Citizendium, the Citizens' Compendium
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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.