Decoding of Projective Reed-Muller Codes by Dividing a Projective Space into Affine Spaces

TL;DR

This paper introduces a decoding algorithm for projective Reed-Muller codes that leverages dividing the projective space into affine spaces, improving decoding efficiency and error correction capabilities.

Contribution

The study presents a novel decoding algorithm for all PRM codes based on space division, along with complexity analysis and performance comparison.

Findings

Decoding algorithm effectively corrects errors in PRM codes.

Algorithm's computational complexity is explicitly determined.

Codeword error rate improves over traditional minimum distance decoding.

Abstract

A projective Reed-Muller (PRM) code, obtained by modifying a (classical) Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The minimum distance and dual code of a PRM code are known, and some decoding examples have been represented for low-dimensional projective space. In this study, we construct a decoding algorithm for all PRM codes by dividing a projective space into a union of affine spaces. In addition, we determine the computational complexity and the number of errors correctable of our algorithm. Finally, we compare the codeword error rate of our algorithm with that of minimum distance decoding.