Crossed Products and Coding Theory

TL;DR

This paper classifies the ambient spaces of crossed product codes up to Hamming-isometry, providing criteria for isometry and exploring examples involving cyclic and elementary abelian groups over various fields.

Contribution

It introduces a classification criterion for crossed product codes based on G-automorphism actions on cohomology, unifying various code families under a common framework.

Findings

Classification of crossed product codes up to Hamming-isometry.

Criteria for isometry involving G-automorphisms and cohomology.

Examples of crossed products over finite fields and complex numbers.

Abstract

Families of codes such as group codes, constacyclic and skew cyclic codes, some of which independently suggested in the literature, turn out to be special instances of the general family of crossed product codes. Hamming-metric is a main feature of ambient code spaces which is used to evaluate the efficiency of their various codes. This note aims at classifying the ambient spaces of crossed products up to Hamming-isometry. We establish a criterion for two crossed products of a group G over a base ring R to be isometric in terms of a certain G-automorphism action on the second cohomology of G with coefficients in R*. This classification is demonstrated by two families of examples, namely crossed products of cyclic groups over finite fields, and twisted group algebras of elementary abelian groups over the complex field and over finite fields. We also determine when crossed products belonging to these families are (relative) semisimple, and in particular, when they admit only trivial codes.