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