The weight distributions of a class of cyclic codes II

TL;DR

This paper extends the analysis of weight distributions for a specific class of cyclic codes by employing number theory tools and elliptic curve point counting, including cases where the field characteristic is 2.

Contribution

It solves a new special case of weight distribution for cyclic codes using character sums and elliptic curve point counting methods.

Findings

Derived weight distributions for the duals of cyclic codes with two zeros

Utilized character sums, Gauss sums, and Jacobi sums in the analysis

Addressed the case where the finite field characteristic is 2

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 use the method developed before 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. The problem of finding the weight distribution is transformed into a problem of evaluating certain character sums over finite fields, which turns out to be associated with counting the number of points on some elliptic curves over finite fields. We also treat the special case that the characteristic of the finite field is 2.