A transform theory for a class of group-invariant codes

A binary cyclic code of length of n can be defined by a set of parity-check equations that is invariant under the additive cyclic group generated by A:i→i+1 modulo n. A class of quasicyclic codes that have parity-check equations invariant under the group generated by a subgroup of A and a subgroup of the multiplicative group M:i →2^ki modulo n is studied in detail. The Fourier transform transforms the action of the additive group, and a linearized polynomial transform transforms the action of the multiplicative group. The form of the transformed code equations permits a BCH-like bound on the minimum distance of the code. The techniques are illustrated by the construction of several codes that are decodable by the Berlekamp-Massey algorithm and that have parameters that approach or surpass those of the best codes known