Alternant code

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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