Evaluation of the Hamming weights of a class of linear codes based on Gauss sums

TL;DR

This paper uses Gauss sums to analyze the Hamming weights of certain linear codes, deriving bounds, weight distributions, and identifying new optimal codes with potential applications in security and data storage.

Contribution

It introduces a novel approach using Gauss sums to evaluate weights and distributions of specific linear codes, including the discovery of new optimal codes.

Findings

Derived lower bounds for minimum Hamming distance.

Evaluated weight distributions in special cases using semi-primitive Gauss sums.

Identified new one-weight, two-weight optimal linear codes.

Abstract

Linear codes with a few weights have been widely investigated in recent years. In this paper, we mainly use Gauss sums to represent the Hamming weights of a class of $q$-ary linear codes under some certain conditions, where $q$ is a power of a prime. The lower bound of its minimum Hamming distance is obtained. In some special cases, we evaluate the weight distributions of the linear codes by semi-primitive Gauss sums and obtain some one-weight, two-weight linear codes. It is quite interesting that we find new optimal codes achieving some bounds on linear codes. The linear codes in this paper can be used in secret sharing schemes, authentication codes and data storage systems.