On the Distance Distribution of Reed-Muller Codes

TL;DR

This paper provides new error bounds for the distance distribution of Reed-Muller codes by counting multivariate polynomials with specific properties, using a character sum method that unifies polynomial enumeration over finite fields.

Contribution

It introduces a novel character sum framework to analyze polynomial counts and systematically studies the coset weight distribution of Reed-Muller codes over large finite fields.

Findings

Established error bounds for Reed-Muller code distance distribution

Developed a unified character sum approach for polynomial enumeration

First systematic study of coset weight distribution for Reed-Muller codes

Abstract

In this paper, we give error bounds for the distance distribution of Reed-Muller codes, extending prior work on the distance distribution of Reed-Solomon codes. This is equivalent to the problem of counting multivariate polynomials over a finite field with prescribed degree, coefficients, and number of zeroes. We provide a solution to this problem using the character sum method, which offers a new unified framework applicable to a broad class of polynomial enumeration problems over finite fields that involve prescribed evaluation vectors. This work effectively makes the first systematic attempt to study the coset weight distribution problem for Reed-Muller codes of fixed degree over large finite fields, which was proposed in MacWilliams and Sloane's 1977 textbook \emph{The Theory of Error Correcting Codes}.