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 finite group. In these realizations, the group algebra is identified with the ambient space, and the group elements with the coordinates of the ambient space. It is an obvious consequence of the definition that every p r -ary affine-invariant code of length p m , with p prime, can be realized as an ideal of the group algebra F p r [ ( F p m , + ) ] , where ( F p m , + ) is the underlying additive group of the field F p m with p m elements. In this paper we describe all the group code structures of an affine-invariant code of length p m in terms of a family of maps from F p m to the group of automorphisms of ( F p m , + ) . We also present a family of non-obvious group code structures in an arbitrary affine-invariant code.