Optimal linear codes with few weights from simplicial complexes

TL;DR

This paper constructs four new families of optimal linear codes over finite fields using simplicial complexes with a single maximal element, determining their parameters and weight distributions, and obtaining optimal codes over finite fields via Gray maps.

Contribution

It introduces four new families of linear codes from simplicial complexes with one maximal element, generalizing previous results and fully determining their parameters and weight distributions.

Findings

Constructed four families of linear codes over rings from simplicial complexes.

Determined parameters and Lee weight distributions of these codes.

Obtained several classes of optimal linear codes over finite fields via Gray map.

Abstract

Recently, constructions of optimal linear codes from simplicial complexes have attracted much attention and some related nice works were presented. Let $q$ be a prime power. In this paper, by using the simplicial complexes of ${\mathbb F}_{q}^m$ with one single maximal element, we construct four families of linear codes over the ring ${\mathbb F}_{q}+u{\mathbb F}_{q}$ ($u^2=0$), which generalizes the results of [IEEE Trans. Inf. Theory 66(6):3657-3663, 2020]. The parameters and Lee weight distributions of these four families of codes are completely determined. Most notably, via the Gray map, we obtain several classes of optimal linear codes over ${\mathbb F}_{q}$, including (near) Griesmer codes and distance-optimal codes.