Improved error bounds for the distance distribution of Reed-Solomon codes

TL;DR

This paper improves error bounds for the distance distribution of Reed-Solomon codes using generating functions, leading to better estimates on polynomial counts and new insights into deep hole classification.

Contribution

It introduces a generating function approach to derive simpler factorial moments and tighter bounds, advancing understanding of Reed-Solomon code properties.

Findings

Enhanced upper bounds for polynomial counting formulas

New classification results for deep holes in Reed-Solomon codes

Improved estimates on the number of polynomials with prescribed properties

Abstract

We use the generating function approach to derive simple expressions for the factorial moments of the distance distribution over Reed-Solomon codes. We obtain better upper bounds for the error term of a counting formula given by Li and Wan, which gives nontrivial estimates on the number of polynomials over finite fields with prescribed leading coefficients and a given number of linear factors. This improvement leads to new results on the classification of deep holes of Reed Solomon codes.