Difference between revisions of "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 }}
 
[[Category:Error detection and correction]]
 
[[Category:Finite fields]]
 
[[Category:Coding theory]]
 
 
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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