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

TL;DR

This work constructs five-Lee-weight linear codes over the semilocal ring $\mathcal{R}=\mathbb{F}_{q}+u\mathbb{F}_{q}$ with $u^{2}=1$ via a defining set and Gauss sums, and analyzes their Lee-weight distributions and the complete weight enumerators of their Gray images. It derives explicit Lee-weight distributions for odd and even extension degrees $m$, and provides the complete Hamming-weight enumerator for the Gray image $\phi(\mathcal{C}_{D})$, leveraging character-theoretic tools and Gauss sums. The paper then applies an Ashikhmin–Barg minimality condition to show that nonzero codewords are minimal under certain $m$ (odd $m\ge5$, even $m\ge6$), enabling secret-sharing schemes with desirable access structures. Overall, it extends weight-distribution results to a broader ring setting (generalizing LL19) and offers concrete constructions with cryptographic relevance through secret sharing.

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.

Theorems & Definitions (19)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5