Alternant code

In coding theory, alternant codes form a class of parameterised error-correcting codes which generalise the BCH codes.


An alternant code over GF(q) of length n is defined by a parity check matrix H of alternant form H<sub>i,j</sub> = α<sub>j</sub><sup>i</sup>y<sub>i</sub>, where the α<sub>j</sub> are distinct elements of the extension GF(q<sup>m</sup>), the y<sub>i</sub> are further non-zero parameters again in the extension GF(q<sup>m</sup>) 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