Index bounds for character sums with polynomials over finite fields

TL;DR

This paper establishes new bounds on character sums over finite fields, improving existing bounds for certain polynomials and applying these results to equations and coding theory.

Contribution

It introduces an index bound that enhances the Weil bound for polynomials with various indices, impacting solutions of equations and cyclic code weights.

Findings

Improved bounds for character sums of polynomials with small and large indices.

Applications to solutions of Artin-Schreier equations.

Results on minimum weights of cyclic codes.

Abstract

We provide an index bound for character sums of polynomials over finite fields. This improves the Weil bound for high degree polynomials with small indices, as well as polynomials with large indices that are generated by cyclotomic mappings of small indices. As an application, we also give some general bounds for numbers of solutions of some Artin-Schreier equations and mininum weights of some cyclic codes.