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On the Distance Distribution of Reed-Muller Codes

Neil Kolekar

TL;DR

This work addresses the distance distribution (and coset weight distribution) for Reed-Muller codes $\,\mathrm{RM}_q(d,v)$ over large finite fields by translating the problem into counting multivariate polynomials with degree constraints and prescribed zero patterns, framed via the reduced polynomial $\,\overline{f-g}\,$ in a truncated polynomial-quotient ring. The authors extend Li–Wan’s character-sum methodology to the multivariate setting, using Lagrange interpolation to describe vanishing sets, Gauss sums on truncated rings, and two key enumerations (ideals and multiplicative characters) to control error terms. The main result is an explicit error bound for $N_k(d,g)$, which yields a sharp approximation to the coset-weight distribution and thus to the distance distribution, generalizing the Reed-Solomon approach to multivariate Reed-Muller codes. The framework provides new tools for polynomial enumeration with prescribed evaluation vectors and zeros, with potential applicability to deeper holes, generalized quotient rings, and related finite-field counting problems with practical coding-theory implications.

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}.

On the Distance Distribution of Reed-Muller Codes

TL;DR

This work addresses the distance distribution (and coset weight distribution) for Reed-Muller codes over large finite fields by translating the problem into counting multivariate polynomials with degree constraints and prescribed zero patterns, framed via the reduced polynomial in a truncated polynomial-quotient ring. The authors extend Li–Wan’s character-sum methodology to the multivariate setting, using Lagrange interpolation to describe vanishing sets, Gauss sums on truncated rings, and two key enumerations (ideals and multiplicative characters) to control error terms. The main result is an explicit error bound for , which yields a sharp approximation to the coset-weight distribution and thus to the distance distribution, generalizing the Reed-Solomon approach to multivariate Reed-Muller codes. The framework provides new tools for polynomial enumeration with prescribed evaluation vectors and zeros, with potential applicability to deeper holes, generalized quotient rings, and related finite-field counting problems with practical coding-theory implications.

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}.
Paper Structure (12 sections, 17 theorems, 96 equations)

This paper contains 12 sections, 17 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Motivating problem
  3. Past work
  4. Preliminaries
  5. Character Sum Setup
  6. Center of approximation
  7. Gauss Sum Computation
  8. Key Enumerations
  9. Distribution of codimension of ideals
  10. Enumerating multiplicative characters
  11. Li-Wan Sieve Application and Proof of Main Result
  12. Future Directions

Key Result

Theorem 1.3

Let $N_k(d, g)$ denote the number of polynomials $f(x_1, \dots, x_v) \in \mathbb{F}_q[x_1, \dots, x_v]$ such that the degree of every variable $x_i$ in $f$ is at most $q-1$, $f-g$ has no nonzero terms of degree $\ell-1$ or lower, and $\overline{f-g}$ has exactly $q^v-k-1$ zeroes. Furthermore, let Then, $N_k(d, g)$ satisfies the error bound where $q_1 = \max(\ell-1, (q-1)/\sqrt{q(p-1)})$ and

Theorems & Definitions (49)

  • Theorem 1.3
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • Proposition 3.5
  • proof
  • Claim 4.1
  • ...and 39 more