List Decoding Quotient Reed-Muller Codes

TL;DR

This work formalizes the notion of ${\tilde{X}}$-quotient Reed-Muller codes, where induced words on a subset ${\tilde{X}}$ satisfy degree-$d$ constraints, and studies how the original RM code's distance and list-decoding radius can be inherited by the quotient code. The authors introduce ${\tilde{X}}$-measurability and a relative regularization framework to handle non-unique lifts from ${\tilde{X}}$ to ${\mathbb{F}}^n$, enabling a transfer of high-rank, equidistribution properties from the ambient space to the subset. The main results show that for constant field size and degree $d<|\mathbb{F}|$, if ${\tilde{X}}$ is a lift-enabler and satisfies a $d$-relative rank-bias property, then RM codes over ${\tilde{X}}$ preserve both distance and list-decoding radius from the full RM code; this extends to high-rank varieties via known additive combinatorics and algebraic-geometry results. Overall, the paper provides a robust framework to analyze quotient codes on subsets, with potential applications to PCPs, puncturing analogues, and new code-construction paradigms in theoretical CS and coding theory.

Abstract

Reed-Muller codes consist of evaluations of $n$-variate polynomials over a finite field $\mathbb{F}$ with degree at most $d$. Much like every linear code, Reed-Muller codes can be characterized by constraints, where a codeword is valid if and only if it satisfies all \emph{degree-$d$} constraints. For a subset $\tilde{X} \subseteq \mathbb{F}^n$, we introduce the notion of \emph{$\tilde{X}$-quotient} Reed-Muller code. A function $F : \tilde{X} \rightarrow \mathbb{F}$ is a valid codeword in the quotient code if it satisfies all the constraints of degree-$d$ polynomials \emph{lying in $\tilde{X}$}. This gives rise to a novel phenomenon: a quotient codeword may have \emph{many} extensions to original codewords. This weakens the connection between original codewords and quotient codewords which introduces a richer range of behaviors along with substantial new challenges. Our goal is to answer the following question: what properties of $\tilde{X}$ will imply that the quotient code inherits its distance and list-decoding radius from the original code? We address this question using techniques developed by Bhowmick and Lovett [BL14], identifying key properties of $\mathbb{F}^n$ used in their proof and extending them to general subsets $\tilde{X} \subseteq \mathbb{F}^n$. By introducing a new tool, we overcome the novel challenge in analyzing the quotient code that arises from the weak connection between original and quotient codewords. This enables us to apply known results from additive combinatorics and algebraic geometry [KZ18, KZ19, LZ21] to show that when $\tilde{X}$ is a \emph{high rank variety}, $\tilde{X}$-quotient Reed-Muller codes inherit the distance and list-decoding parameters from the original Reed-Muller codes.

Theorems & Definitions (106)

• Definition 1.1: The ${\tilde{X}}$-Induced Code
• Definition 1.2: The ${\tilde{X}}$-Quotient Code
• Theorem : List Decoding Quotient Reed-Muller Code
• Corollary : List Decoding Quotient Reed-Muller Code: High Rank Variety
• Theorem 1.4
• Definition 1.5: Measurable
• Definition 1.6: Relative Rank, informal. See definition \ref{['def:relative-rank-of-polynomial']}
• Definition 1.7: ${\tilde{X}}$-measurable, Incomplete Definition
• Definition 1.8: ${\tilde{X}}$-measurable
• Theorem 1.9: Relative Regularization Process, Informal, See Theorem \ref{['theorem:regularization-in-X']}