Group code structures of affine-invariant codes

Abstract

A group code structure of a linear code is a description of the code as one-sided or two-sided ideal of a group algebra of a nite group. In these realizations, the group algebra is identi ed with the ambient space, and the group elements with the coordinates of the ambient space. It is an obvious consequence of the de nition that every pr-ary a ne-invariant code of length pm, with p prime, can be realized as an ideal of the group algebra Fpr [(Fpm; +)], where (Fpm; +) is the underlying additive group of the eld Fpm with pm elements. In this paper we describe all the group code structures of an a ne-invariant code of length pm in terms of a family of maps from Fpm to the group of automorphisms of (Fpm; +). We also present a familly of non-obvious group code structures in an arbitrary a ne-invariant code.