A class of few-Lee weight $\mathbb{Z}_2[u]$-linear codes using simplicial complexes and minimal codes via Gray map

TL;DR

This paper constructs a new class of few-Lee weight linear codes over a mixed alphabet ring using simplicial complexes, analyzes their weight distribution, and demonstrates their applications in secret sharing schemes.

Contribution

It introduces a novel construction of few-Lee weight codes over  with simplicial complexes and shows their Gray images are minimal and self-orthogonal.

Findings

Determined the Lee weight distribution of the codes.

Proved Gray images are self-orthogonal.

Established an infinite family of minimal codes for secret sharing.

Abstract

Recently some mixed alphabet rings are involved in constructing few-Lee weight additive codes with optimal or minimal Gray images using suitable defining sets or down-sets. Inspired by these works, we choose the mixed alphabet ring $\mathbb{Z}_2\mathbb{Z}_2[u]$ to construct a special class of linear code $C_L$ over $\mathbb{Z}_2[u]$ with $u^2=0$ by employing simplicial complexes generated by a single maximal element. We show that $C_L$ has few-Lee weights by determining the Lee weight distribution of $C_L$. Theoretically, this shows that we may employ simplicial complexes to obatin few-weight codes even in the case of mixed alphabet rings. We show that the Gray image of $C_L$ is self-orthogonal and we have an infinite family of minimal codes over $\mathbb{Z}_2$ via Gray map, which can be used to secret sharing schemes.