The Generalized Reed-Muller codes in a modular group algebra

TL;DR

This paper investigates properties of modular group algebras over Galois rings and provides a new description of Generalized Reed-Muller codes within this algebraic framework.

Contribution

It introduces a novel description of Generalized Reed-Muller codes in the context of modular group algebras over Galois rings.

Findings

Properties of modular group algebra $\

Description of Generalized Reed-Muller codes in $\

New algebraic framework for coding theory analysis

Abstract

First we study some properties of the modular group algebra $\mathbb{F}_{p^r}[G]$ where $G$ is the additive group of a Galois ring of characteristic $p^r$ and $\mathbb{F}_{p^r}$ is the field of $p^r$ elements. Secondly a description of the Generalized Reed-Muller codes over $\mathbb{F}_{p^r}$ in $\mathbb{F}_{p^r}[G]$ is presented.