On abelian and cyclic group codes

TL;DR

This paper characterizes abelian and cyclic group codes, establishing conditions on their minimum Hamming weight and showing their equivalence to certain subspaces of repeated codes, with detailed automorphism descriptions.

Contribution

It provides a condition on the minimum Hamming weight of abelian group codes and characterizes their structure and automorphisms, also extending results to cyclic group codes.

Findings

Any special abelian group code is permutationally equivalent to a subspace of repeated codes.

Complete characterization of permutation automorphisms of these codes.

Equivalent characterization of cyclic group codes.

Abstract

We determine a condition on the minimum Hamming weight of some special abelian group codes and, as a consequence of this result, we establish that any such code is, up to permutational equivalence, a subspace of the direct sum of $s$ copies of the repetition code of length $t$, for some suitable positive integers $s$ and $t$. Moreover, we provide a complete characterisation of permutation automorphisms of the linear code $C=\bigoplus_{i=1}^{s}Rep_{t}(\mathbb{F}_{q})$ and we establish that such a code is an abelian group code, for every pair of integers $s,t\geq1$. Finally, in a similar fashion as for abelian group codes, we give an equivalent characterisation of cyclic group codes.