Group Algebra and Coding Theory

TL;DR

This paper reviews the use of group algebras in coding theory, highlighting their role in constructing minimal codes and analyzing their properties, based on recent research developments.

Contribution

It summarizes recent advances in applying group algebra techniques to coding theory, focusing on idempotent computation and minimal code analysis.

Findings

Group algebra techniques facilitate the construction of minimal codes.

Idempotent elements in group algebras are key to understanding code structure.

Recent research has expanded the application of algebraic methods in coding theory.

Abstract

Group algebras have been used in the context of Coding Theory since the beginning of the latter, but not in its full power. The work of Ferraz and Polcino Milies entitled Idempotents in group algebras and minimal abelian codes (Finite Fields and their Applications, 13, (2007) 382-393) gave origin to many thesis and papers linking these two subjects. In these works, the techniques of group algebras are mainly brought into play for the computing of the idempotents that generate the minimal codes and the minimum weight of such codes. In this paper I summarize the main results of the work done by doctorate students and research partners of Polcino Milies and Ferraz.