On the Category of Group Codes

TL;DR

This paper studies the structure of group codes, showing that certain classes are indecomposable and that all group codes can be decomposed into indecomposables, with implications for automorphism groups and cyclic codes.

Contribution

It introduces a decomposition theorem for group codes into indecomposable components and characterizes automorphism groups based on this decomposition.

Findings

Every non-trivial MDS, perfect, and constant weight nondegenerate group code is indecomposable.

Every group code can be expressed as a direct sum of indecomposable group codes.

The structure of decomposable cyclic group codes is explicitly determined.

Abstract

For the category of group codes, that generalizes the category of linear codes over a finite field, and with the generalized notions of direct sums and ndecomposable group codes, we prove that every MDS non trivial code, every perfect non trivial code, and every constant weight nondegenerate group code are indecomposable. We prove that every group code is a direct sum of indecomposable group codes, and using this result we obtain the automorphism groups of any group code in terms of its decomposition in indecomposable components. We conclude with the determination of the structure of decomposable cyclic group codes.