Counting polynomials over finite fields with prescribed leading coefficients and linear factors

TL;DR

This paper develops methods to count polynomials over finite fields with specific leading coefficients and linear factors, which has applications in coding theory, particularly in analyzing Reed-Solomon codes.

Contribution

The paper introduces a generating function approach combined with sieve techniques to derive explicit formulas for counting such polynomials, extending previous results.

Findings

Derived explicit formulas for counting polynomials with prescribed properties

Extended previous results by Li, Wan, Zhou, Wang, and Wang

Simplified expressions in special cases using sieve methods

Abstract

We count the number of polynomials over finite fields with prescribed leading coefficients and a given number of linear factors. This is equivalent to counting codewords in Reed-Solomon codes which are at a certain distance from a received word. We first apply the generating function approach, which is recently developed by the author and collaborators, to derive expressions for the number of monic polynomials with prescribed leading coefficients and linear factors. We then apply Li and Wan's sieve formula to simplify the expressions in some special cases. Our results extend and improve some recent results by Li and Wan, and Zhou, Wang and Wang.