The Generalized Reed-Muller codes and the radical powers of a modular algebra
Harinaivo Andriatahiny
TL;DR
This paper provides a new proof of the characterization of Reed-Muller codes as radical powers in modular algebras and extends the approach to generalized codes over non-prime fields.
Contribution
It introduces a novel proof technique for Reed-Muller codes and generalizes the framework to non-prime fields.
Findings
Reed-Muller codes are characterized as radical powers in modular algebras.
The proof method is extended to generalized Reed-Muller codes over non-prime fields.
The approach offers new insights into the algebraic structure of these codes.
Abstract
First, a new proof of Berman and Charpin's characterization of the Reed-Muller codes over the binary field or over an arbitrary prime field is presented. These codes are considered as the powers of the radical of a modular algebra. Secondly, the same method is used for the study of the Generalized Reed-Muller codes over a non prime field.
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Taxonomy
TopicsCoding theory and cryptography · Islamic Finance and Communication · Cooperative Communication and Network Coding