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.

翻译：近年来，从单纯复形构造最优线性码的方法引起了广泛关注，并已取得若干优秀成果。令$q$为素数幂。本文利用${\mathbb F}_{q}^m$上具有单个极大元的单纯复形，在环${\mathbb F}_{q}+u{\mathbb F}_{q}$（$u^2=0$）上构造了四族线性码，推广了文献[IEEE Trans. Inf. Theory 66(6):3657-3663, 2020]的结果。完整确定了这四族码的参数与Lee重量分布。特别重要的是，通过Gray映射，我们得到了${\mathbb F}_{q}$上的多类最优线性码，包括（近似）Griesmer码和距离最优码。