Distance Distribution to Received Words in Reed-Solomon Codes

TL;DR

This paper derives bounds on the number of low-degree polynomials that, when added to a fixed polynomial, have a specified number of roots over a finite field, with applications to Reed-Solomon code list decoding.

Contribution

It extends previous explicit formulas to broader cases, providing bounds on polynomial root distributions and an asymptotic estimate for Reed-Solomon list sizes.

Findings

Bounds on polynomial root counts for general degrees

Asymptotic formula for Reed-Solomon list size

Extension of explicit formulas beyond known cases

Abstract

Let $\mathbb{F}_q$ be the finite field of $q$ elements. In this paper we obtain bounds on the following counting problem: given a polynomial $f(x)\in \mathbb{F}_q[x]$ of degree $k+m$ and a non-negative integer $r$, count the number of polynomials $g(x)\in \mathbb{F}_q[x]$ of degree at most $k-1$ such that $f(x)+g(x)$ has exactly $r$ roots in $\mathbb{F}_q$. Previously, explicit formulas were known only for the cases $m=0, 1, 2$. As an application, we obtain an asymptotic formula on the list size of the standard Reed-Solomon code $[q, k, q-k+1]_q$.