The List Decoding Radius for Reed–Muller Codes Over Small Fields

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 $\text {RM}_{ {\mathbb {F}}}(n,d)$ is defined by $n$ -variate degree- $d$ polynomials over ${\mathbb {F}}$ . In this paper, 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 ${\delta }(d)$ the normalized minimal distance of $\text {RM}_{ {\mathbb {F}}}(n,d)$ , then the number of codewords in any ball of radius ${\delta }(d)- {\varepsilon }$ is bounded by $c=c(p,d, {\varepsilon })$ independent of $n$ . This resolves a conjecture of Gopalan et al. , who among other results proved it in the special case of ${\mathbb {F}}= {\mathbb {F}}_{2}$ ; and extends the work of Gopalan 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 $\text {RM}_{ {\mathbb {F}}}(n,d)$ in a ball of radius ${\delta }(e) - {\varepsilon }$ 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 et al. 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 the compositions of polynomials.