Five-Lee-weight linear codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}$

TL;DR

This paper constructs five-Lee-weight linear codes over a semi-local ring using Gauss sums, derives their weight enumerators, and demonstrates their application in secret sharing schemes.

Contribution

It introduces a new class of linear codes over a semi-local ring with explicit weight distributions and applications to secret sharing.

Findings

Constructed five-Lee-weight linear codes over _{q}+u_{q}.

Derived complete Hamming-weight enumerators for the codes.

Applied the codes to secret sharing schemes.

Abstract

In this study, linear codes having their Lee-weight distributions over the semi-local ring $\mathbb{F}_{q}+u\mathbb{F}_{q}$ with $u^{2}=1$ are constructed using the defining set and Gauss sums for an odd prime $q $. Moreover, we derive complete Hamming-weight enumerators for the images of the constructed linear codes under the Gray map. We finally show an application to secret sharing schemes.