Group code structures on affine-invariant codes

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 well known that every affine-invariant code of length pm, with p prime, can be realized as an ideal of the group algebra FI, where I is the underlying additive group of the field with pm elements. In this paper we describe all the group code structures of an affine-invariant code of length pm in terms of a family of maps from I to the group of automorphisms of I. Affine-invariant codes were firstly introduced by Kasami, Lin and Peterson [KLP2] as a generalization of Reed-Muller codes. This class of codes has received the attention of several authors because of its good algebraic and decoding properties [D, BCh, ChL, Ho, Hu]. The length of an affine-invariant code is a prime power p, where p is the characteristic of the finite field F which plays the role of alphabet. It is well known that every affine-invariant code of length p over the field F can be realized as an ideal of the group algebra FI, where I is the underlying additive group of the field with p elements, i.e. I is the elementary abelian group of order p. In this realization, the group elements are identified with the elements of the standard base of the ambient space F m . We refer to these realizations of codes as one-sided or two-sided ideals in group algebras as group code structures of the given code. In this paper we study all the possible group code structures on an affine-invariant code. Our main tools are an intrinsical characterization of group codes obtained in [BRS] and a description of the group of permutation automorphisms of nontrivial affine-invariant codes given in [BCh]. These results are reviewed in Section 1, where we also recall the definition and main properties of affine-invariant codes. In Section 2, we describe all the group code structures of an affineinvariant code C in terms of a family of maps I → Ga,b where Ga,b is a subgroup of the group of automorphism of I depending on two integers a and b which ∗ Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia. Spain. email: [email protected], [email protected], [email protected] Partially supported by D.G.I. of Spain and Fundación Séneca of Murcia.