In coding theory, alternant codes form a class of parameterised

error-correcting codes In computing, telecommunication, information theory, and coding theory, an error correction code, sometimes error correcting code, (ECC) is used for controlling errors in data over unreliable or noisy communication channels. The central idea is ...

which generalise the

BCH code In coding theory, the Bose–Chaudhuri–Hocquenghem codes (BCH codes) form a class of cyclic error-correcting codes that are constructed using polynomials over a finite field (also called ''Galois field''). BCH codes were invented in 1959 ...

Definition

An ''alternant code'' over GF(''q'') of length ''n'' is defined by a parity check matrix ''H'' of alternant form ''H''_''i'',''j'' = α_jⁱ''y''_''i'', where the α_''j'' are distinct elements of the extension GF(''q''^''m''), the ''y''_''i'' are further non-zero parameters again in the extension GF(''q''^''m'') 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'' − ''m''δ and minimum distance ≥ δ + 1. There exist long alternant codes which meet the

Gilbert–Varshamov bound In coding theory, the Gilbert–Varshamov bound (due to Edgar Gilbert and independently Rom Varshamov.) is a limit on the parameters of a (not necessarily linear) code. It is occasionally known as the Gilbert– Shannon–Varshamov bound (or the GSV ...

. The class of alternant codes includes *

s * Goppa codes *

Srivastava code In coding theory, Srivastava codes, formulated by Professor J. N. Srivastava, form a class of parameterised error-correcting codes which are a special case of alternant code In coding theory, alternant codes form a class of parameterised error-cor ...

References

* Error detection and correction Finite fields Coding theory {{crypto-stub