# Alternant code

This is the approved revision of this page, as well as being the most recent.In coding theory, **alternant codes** form a class of parameterised error-correcting codes which generalise the BCH codes.

## DefinitionEdit

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*.

## PropertiesEdit

The parameters of this alternant code are length *n*, dimension ≥ *n* − *m*δ and minimum distance ≥ δ + 1.
There exist long alternant codes which meet the Gilbert-Varshamov bound.

The class of alternant codes includes