High Rate Multivariate Polynomial Evaluation Codes

TL;DR

This paper breaks the rate barrier for multivariate polynomial evaluation codes by constructing two explicit high-rate families, CAP and GAP, whose evaluation domains preserve constant relative distance for fixed dimension $m$. CAP uses simplex-based combinatorial shapes, while GAP uses intersections of hyperplanes in general position, enabling rate $R \ge 1-\varepsilon$ and relative distance $\Omega(1)$ (for constant $m$) with efficient decoding. A central technical tool is a generalized Schwartz–Zippel lemma and a suite of structured zero-pattern analyses that tie zero sets to robust distance bounds, supporting both unique decoding up to half the minimum distance and local testability (notably for GAP). The GAP construction, in particular, yields a Tanner-code–like structure with local testability and a clean concatenation-based decoding framework, while CAP provides a near-RM analogue with strong decoding by peeling/GMD techniques adapted to variable inner code lengths. Altogether, the work delivers practical high-rate multivariate codes with efficient decoding and robust local testing, broadening the toolkit for PCPs, derandomization, and algebraic complexity applications.

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 $Ω(1)$ and rate $1 - ε$ for any $ε> 0$. 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.

Figures (5)

• Figure 1: Possible locations of zeros of a bivariate polynomial, $f(X, Y)$ of degree $10$, $Y$-degree $3$, evaluated on $\mathbb{N}_{< 15} \times \mathbb{N}_{< 15}$. Zeros are denoted with black squares.
• Figure 2: A possible zero pattern (same as \ref{['fig:zero-pattern']}) overlayed with the triangle evaluation set from \ref{['example:triangle']} in orange.
• Figure 3: Example of shifting the zeros from \ref{['fig:triangle']}
• Figure 4: Comparing rate vs. distance trade-offs for the grid, simplex and step evaluation sets
• Figure 5: Comparing rate vs. robustness trade-off of the simplex construction to the upper bound

Theorems & Definitions (117)

• definition 2.1: Polynomial Evaluation Code
• definition 2.2: Reed-Solomon Code
• definition 2.3: Reed-Muller Code
• definition 2.4: Hitting sets
• definition 2.5: Interpolating sets
• definition 2.6
• theorem 2.7: Forney forneyGeneralizedMinimumDistance1966
• lemma 3.1
• proof
• claim 3.2