The Complete Weight Enumerator of Several Cyclic Codes

TL;DR

This paper introduces a general method to compute the complete weight enumerator of cyclic codes using exponential sum value distributions, and applies it to specific p-ary cyclic codes, providing explicit examples.

Contribution

A novel strategy for calculating the complete weight enumerator of cyclic codes based on exponential sum distributions, with concrete applications to p-ary codes.

Findings

Derived the complete weight enumerator for several p-ary cyclic codes

Established a link between weight enumerators and exponential sum distributions

Provided explicit examples illustrating the method's effectiveness

Abstract

Cyclic codes have attracted a lot of research interest for decades. In this paper, for an odd prime $p$, we propose a general strategy to compute the complete weight enumerator of cyclic codes via the value distribution of the corresponding exponential sums. As applications of this general strategy, we determine the complete weight enumerator of several $p$-ary cyclic codes and give some examples to illustrate our results.