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

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.

12 pages, size of code is changed in Theorem 3.2, proposition 4.1 is added, some examples are added in both section 3 and 4