Points-Polynomials Incidence Theorem with Applications to Coding Theory

TL;DR

This paper extends incidence bounds over finite fields to higher-degree polynomials, employing matrix spectral analysis, and applies these results to demonstrate new average-radius list-decodability properties of Reed-Solomon codes.

Contribution

It introduces a novel incidence bound for points and polynomials over finite fields and applies it to establish average-radius list-decodability of Reed-Solomon codes.

Findings

Established a bound on point-polynomial incidences in finite fields.

Proved Reed-Solomon codes are average-radius list-decodable with constant list size.

Connected incidence geometry with coding theory applications.

Abstract

This paper focuses on incidences over finite fields, extending to higher degrees a result by Vinh \cite{VINH20111177} on the number of point-line incidences in the plane $\mathbb{F}^2$, where $\mathbb{F}$ is a finite field. Specifically, we present a bound on the number of incidences between points and polynomials of bounded degree in $\mathbb{F}^2$. Our approach employs a singular value decomposition of the incidence matrix between points and polynomials, coupled with an analysis of the related group algebras. This bound is then applied to coding theory, specifically to the problem of average-radius list decoding of Reed-Solomon (RS) codes. We demonstrate that RS codes of certain lengths are average-radius list-decodable with a constant list size, which is dependent on the code rate and the distance from the Johnson radius. While a constant list size for list-decoding of RS codes in this regime was previously established, its existence for the stronger notion of average-radius list-decoding was not known to exist.