High Rate Multivariate Polynomial Evaluation Codes

2024-10-17

Electron. Colloquium Comput. Complex.

AI Summary

This work addresses the low-rate limitation (≤1/m!) of classical Reed–Muller codes by introducing two families of high-rate multivariate polynomial evaluation codes that break the traditional rate–distance trade-off. The core method evaluates low-degree polynomials over carefully constructed subsets S ⊆ 𝔽_q^m: one family leverages simplex-based geometric structures, while the other imposes algebraic constraints to ensure local testability. Both constructions achieve rate 1−ε (for any ε > 0) and Ω(1) relative minimum distance in constant dimension m, marking the first result attaining asymptotically optimal rate without sacrificing distance in constant-dimensional settings. For decoding, we design polynomial-time algorithms combining list decoding and majority voting, enabling correction of up to half the minimum distance. Moreover, the second construction is rigorously proven to be locally testable.

Technology Category

• Machine Learning: Multi-instance/Multi-view Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization

Application Category

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

Abstract

The classical Reed-Muller codes over a finite field $mathbb{F}_q$ are based on evaluations of $m$-variate polynomials of degree at most $d$ over a product set $U^m$, for some $d$ less than $|U|$. Because of their good distance properties, as well as the ubiquity and expressive power of polynomials, these codes have played an influential role in coding theory and complexity theory. This is especially so in the setting of $U$ being ${mathbb{F}}_q$ where they possess deep locality properties. However, these Reed-Muller codes have a significant limitation in terms of the rate achievable -- the rate cannot be more than $frac{1}{m{!}} = exp(-m log m)$. In this work, we give the first constructions of multivariate polynomial evaluation codes which overcome the rate limitation -- concretely, we give explicit evaluation domains $S subseteq mathbb{F}_q^m$ on which evaluating $m$-variate polynomials of degree at most $d$ gives a good code. For $m= O(1)$, these new codes have relative distance $Omega(1)$ and rate $1 - epsilon$ for any $epsilon>0$. In fact, we give two quite different constructions, and for both we develop efficient decoding algorithms for these codes that can decode from half the minimum distance. The first of these codes is based on evaluating multivariate polynomials on simplex-like sets whereas the second construction is more algebraic, and surprisingly (to us), has some strong locality properties, specifically, we show that they are locally testable.

Problem

• Reed-Muller Codes
• Information Capacity
• Error Correction

Innovation

• Reed-Muller Codes
• Triangular Mathematical Structure
• Local Verification Method