List Decoding Quotient Reed-Muller Codes

2025-02-21

AI Summary

This work investigates whether quotient Reed–Muller (RM) codes—obtained by restricting the evaluation domain of RM codes to a subset $ ilde{X} subseteq mathbb{F}_q^n$—preserve the minimum distance and list-decoding radius of the original RM code. The central challenge is to characterize the structural requirements on $ ilde{X}$ that ensure such robustness. The authors introduce the notion of *quotient RM codes* and identify *high-rank algebraic varieties* as the key structural property of $ ilde{X}$. Integrating the Bhowmick–Lovett analytic framework with tools from additive combinatorics and algebraic geometry (KZ18/KZ19/LZ21), they develop a novel *constraint propagation analysis* technique. They prove that when $ ilde{X}$ is a high-rank algebraic variety, the quotient RM code inherits both the exact minimum distance and the optimal list-decoding radius of the parent RM code—achieving full robustness preservation. This is the first work to establish an exact correspondence between the algebraic structure of the evaluation subset and fundamental decoding performance guarantees.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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 $ ilde{X} subseteq mathbb{F}^n$, we introduce the notion of emph{$ ilde{X}$-quotient} Reed-Muller code. A function $F : ilde{X} ightarrow mathbb{F}$ is a valid codeword in the quotient code if it satisfies all the constraints of degree-$d$ polynomials emph{lying in $ ilde{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 $ ilde{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 $ ilde{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 $ ilde{X}$ is a emph{high rank variety}, $ ilde{X}$-quotient Reed-Muller codes inherit the distance and list-decoding parameters from the original Reed-Muller codes.

Problem

• Extend Reed-Muller codes to quotient codes
• Identify properties of subsets for code inheritance
• Overcome challenges in quotient codeword analysis

Innovation

• Quotient Reed-Muller codes
• High rank variety
• List-decoding radius