Codes in algebras of direct products of groups

TL;DR

This paper derives the Wedderburn-Artin decomposition for semisimple group algebras of direct product groups, providing formulas for group codes and their dimensions, with explicit descriptions for certain classes of groups.

Contribution

It offers a new algebraic framework for understanding group codes in direct product group algebras, including explicit formulas and classifications.

Findings

Wedderburn-Artin decomposition for direct product groups

Formulas for counting and dimension of group codes

Complete algebraic descriptions for specific group classes

Abstract

In this paper we obtain the Wedderburn-Artin decomposition of a semisimple group algebra associated to a direct product of finite groups. We also provide formulae for the number of all possible group codes, and their dimensions, that can be constructed in a group algebra. As particular cases, we present the complete algebraic description of the group algebra of any direct product of groups whose direct factors are cyclic, dihedral, or generalised quaternion groups.