Optimal linear codes with few weights from simplicial complexes

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.