Group code structures on affine-invariant codes

TL;DR

This paper characterizes all group code structures of affine-invariant codes of length p^m over finite fields, using maps to automorphisms of the additive group, enhancing understanding of their algebraic structure.

Contribution

It provides a complete description of all group code structures of affine-invariant codes in terms of automorphism maps, extending prior knowledge of their algebraic realizations.

Findings

All group code structures are described via maps from the additive group to automorphisms.

The characterization applies to codes of length p^m over finite fields.

The results unify and extend previous realizations 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 well known that every affine-invariant code of length $p^m$, with $p$ prime, can be realized as an ideal of the group algebra $\F\I$, where $\I$ is the underlying additive group of the field 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 $\I$ to the group of automorphisms of $\I$.