Deep Holes in Reed-Solomon Codes Based on Dickson Polynomials

TL;DR

This paper investigates deep holes in Reed-Solomon codes with evaluation sets derived from Dickson polynomials, providing new methods to identify non-deep holes using character sum estimates and subset sum reductions.

Contribution

It introduces a novel approach to analyze deep holes in Reed-Solomon codes with Dickson polynomial evaluation sets, connecting the problem to subset sum and Waring's problem.

Findings

Identifies conditions under which received words are not deep holes.

Reduces the deep hole problem to a subset sum problem involving Dickson polynomials.

Provides bounds and estimates for character sums over evaluation sets.

Abstract

For an $[n,k]$ Reed-Solomon code $\mathcal{C}$, it can be shown that any received word $r$ lies a distance at most $n-k$ from $\mathcal{C}$, denoted $d(r,\mathcal{C})\leq n-k$. Any word $r$ meeting the equality is called a deep hole. Guruswami and Vardy (2005) showed that for a specific class of codes, determining whether or not a word is a deep hole is NP-hard. They suggested passingly that it may be easier when the evaluation set of $\mathcal{C}$ is large or structured. Following this idea, we study the case where the evaluation set is the image of a Dickson polynomial, whose values appear with a special uniformity. To find families of received words that are not deep holes, we reduce to a subset sum problem (or equivalently, a Dickson polynomial-variation of Waring's problem) and find solution conditions by applying an argument using estimates on character sums indexed over the evaluation set.