The Generalized Reed-Muller codes in a modular group algebra

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.