The weight distributions of a class of cyclic codes

TL;DR

This paper introduces a novel approach using number theory tools to determine the weight distributions of certain cyclic codes, extending previous results by solving an additional special case through character sums and elliptic curve point counting.

Contribution

It presents a new method employing characters, Gauss sums, and elliptic curves to analyze weight distributions of cyclic codes, broadening the scope of prior techniques.

Findings

Successfully determined the weight distribution for a new special case of cyclic codes.

Demonstrated the effectiveness of character sums and elliptic curve point counting in coding theory.

Extended the methodology to potentially analyze other cases of cyclic codes.

Abstract

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases. In this paper we provide a slightly different approach toward the general problem and use it to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums to transform the problem of finding the weight distribution into a problem of evaluating certain character sums over finite fields, which on the special case is related with counting the number of points on some elliptic curves over finite fields. Other cases are also possible by this method.