Several classes of cyclic codes with either optimal three weights or a few weights

TL;DR

This paper introduces new classes of cyclic codes with either exactly three weights or very few weights, using Gauss sums, and demonstrates their optimality and potential applications in coding theory.

Contribution

It presents a general construction of cyclic codes with few weights using Gauss sums and identifies classes that are optimal or have few weights, solving open problems.

Findings

Constructed a class of optimal three-weight cyclic codes achieving the Griesmer bound.

Developed several classes of cyclic codes with only a few weights.

Generalized previous results and solved open problems in cyclic code classification.

Abstract

Cyclic codes with a few weights are very useful in the design of frequency hopping sequences and the development of secret sharing schemes. In this paper, we mainly use Gauss sums to represent the Hamming weights of a general construction of cyclic codes. As applications, we obtain a class of optimal three-weight codes achieving the Griesmer bound, which generalizes a Vega's result in \cite{V1}, and several classes of cyclic codes with only a few weights, which solve the open problem in \cite{V1}.