[Paper Review] List decoding Reed-Muller codes over small fields

Abhishek Bhowmick, Shachar Lovett|arXiv (Cornell University)|Jul 13, 2014

Coding theory and cryptography20 references3 citations

TL;DR

This paper resolves a long-standing conjecture by proving that the list decoding radius of Reed-Muller codes over fixed prime fields equals their minimum distance, establishing that the number of codewords within radius δ(d)−ε is bounded by a constant independent of n. The proof relies on higher-order Fourier analysis, a generalized Schwartz-Zippel lemma, and an extension of the Frieze-Kannan regularity lemma to function spaces.

ABSTRACT

The list decoding problem for a code asks for the maximal radius up to which any ball of that radius contains only a constant number of codewords. The list decoding radius is not well understood even for well studied codes, like Reed-Solomon or Reed-Muller codes. Fix a finite field $\mathbb{F}$. The Reed-Muller code $\mathrm{RM}_{\mathbb{F}}(n,d)$ is defined by $n$-variate degree-$d$ polynomials over $\mathbb{F}$. In this work, we study the list decoding radius of Reed-Muller codes over a constant prime field $\mathbb{F}=\mathbb{F}_p$, constant degree $d$ and large $n$. We show that the list decoding radius is equal to the minimal distance of the code. That is, if we denote by $δ(d)$ the normalized minimal distance of $\mathrm{RM}_{\mathbb{F}}(n,d)$, then the number of codewords in any ball of radius $δ(d)-\varepsilon$ is bounded by $c=c(p,d,\varepsilon)$ independent of $n$. This resolves a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008], who among other results proved it in the special case of $\mathbb{F}=\mathbb{F}_2$; and extends the work of Gopalan [FOCS 2010] who proved the conjecture in the case of $d=2$. We also analyse the number of codewords in balls of radius exceeding the minimal distance of the code. For $e \leq d$, we show that the number of codewords of $\mathrm{RM}_{\mathbb{F}}(n,d)$ in a ball of radius $δ(e) - \varepsilon$ is bounded by $\exp(c \cdot n^{d-e})$, where $c=c(p,d,\varepsilon)$ is independent of $n$. The dependence on $n$ is tight. This extends the work of Kaufman-Lovett-Porat [IEEE Inf. Theory 2012] who proved similar bounds over $\mathbb{F}_2$. The proof relies on several new ingredients: an extension of the Frieze-Kannan weak regularity to general function spaces, higher-order Fourier analysis, and an extension of the Schwartz-Zippel lemma to compositions of polynomials.

Motivation & Objective

• To resolve the conjecture that the list decoding radius of Reed-Muller codes over fixed prime fields equals their minimum distance.
• To extend previous results on list decoding for low-degree polynomials over F₂ and Fₚ to arbitrary fixed prime fields and degrees.
• To analyze the number of codewords within radius exceeding the minimum distance, providing tight bounds in terms of n and degree difference.
• To develop new analytical tools, including a higher-order Fourier analysis framework and an extended Schwartz-Zippel lemma for polynomial compositions.
• To lay the groundwork for extending list decoding results to non-prime fields, identifying the missing theoretical ingredient.

Proposed method

• Extends the Frieze-Kannan weak regularity lemma to general function spaces over finite fields, enabling decomposition of low-degree polynomial functions.
• Applies higher-order Fourier analysis to analyze the structure of low-degree polynomials and their agreement with received words.
• Introduces a generalized Schwartz-Zippel lemma that bounds the number of roots of compositions of polynomials, crucial for controlling list size.
• Uses a recursive decomposition of polynomials into multilinear forms over auxiliary variables to reduce the problem to lower-degree components.
• Employs statistical distance arguments to relate the agreement of a function with a codeword to the agreement of its structured components.
• Combines these tools to bound the number of low-degree polynomials within a given distance from any received word, proving constant list size up to the minimum distance.

Experimental results

Research questions

• RQ1 Is the list decoding radius of Reed-Muller codes over fixed prime fields equal to their minimum distance?
• RQ2 Can the number of degree-d polynomials within radius δ(e)−ε of any received word be bounded by exp(c·n^{d−e}) for e ≤ d and constant c?
• RQ3 What structural properties of low-degree polynomials allow for tight list size bounds beyond the Johnson radius?
• RQ4 Can higher-order Fourier analysis be extended to function spaces over arbitrary finite fields to enable such bounds?
• RQ5 What are the limitations of current techniques when extending to non-prime fields?

Key findings

• The list decoding radius of RM_F(n,d) over F_p equals its minimum distance δ(d), resolving a conjecture by Gopalan-Klivans-Zuckerman for prime fields.
• For any e ≤ d and ε > 0, the number of degree-d polynomials within distance δ(e)−ε of any received word is at most exp(c·n^{d−e}) for a constant c = c(p,d,ε), and this dependence on n is tight.
• The bound on list size is constant when e = d, confirming that list decoding is possible up to the minimum distance for all fixed prime fields and degrees.
• The proof introduces a new generalized Schwartz-Zippel lemma that applies to compositions of polynomials, enabling control over agreement sets.
• A novel extension of the Frieze-Kannan regularity lemma to general function spaces is developed and used to decompose polynomial functions into structured components.
• The analysis shows that the list size grows exponentially in n^{d−e} when the radius exceeds δ(e)−ε, and this growth rate is tight for the given parameters.