Cyclicity of Binary Group Codes

TL;DR

This paper explores binary group codes and shows that some cannot be represented as cyclic codes, expanding the understanding of code structures.

Contribution

The paper introduces ω|ω¯ codes as self-dual group codes and proves their non-equivalence to cyclic codes under specific conditions.

Findings

ω|ω¯ codes are self-dual group codes over the abelian group C2×Ck.

For even k>2, these codes are not permutationally equivalent to cyclic codes.

Computational results show non-cyclic group codes exist beyond cyclic group algebras.

Abstract

In this paper, we study the cyclicity of binary group codes, identifying them as ideals in a group algebra. We focus on the construction of ω|ω¯ codes, proving that they are self-dual group codes over the abelian group C2×Ck. We demonstrate that for even integers k>2, if the polynomial xk−1 splits into self-reciprocal irreducible factors, these codes are not permutationally equivalent to any cyclic code. Additionally, we present computational results for binary group codes of length n<24 using the MAGMA software (V2.29-4). These results confirm that while all cyclic codes in this range are equivalent to abelian group codes, there exist non-cyclic group codes that cannot be realized as ideals in a cyclic group algebra, highlighting the strictly larger scope of the class of group codes.