In this paper, we construct a large family of projective linear codes over ${\mathbb F}_{q}$ from the general simplicial complexes of ${\mathbb F}_{q}^m$ via the defining-set construction, which generalizes the results of [IEEE Trans. Inf. Theory 66(11):6762-6773, 2020]. The parameters and weight distribution of this class of codes are completely determined. By using the Griesmer bound, we give a necessary and sufficient condition such that the codes are Griesmer codes and a sufficient condition such that the codes are distance-optimal. For a special case, we also present a necessary and sufficient condition for the codes to be near Griesmer codes. Moreover, by discussing the cases of simplicial complexes with one, two and three maximal elements respectively, the parameters and weight distributions of the codes are given more explicitly, which shows that the codes are at most $2$-weight, $5$-weight and $19$-weight respectively. By studying the optimality of the codes for the three cases in detail, many infinite families of optimal linear codes with few weights over ${\mathbb F}_{q}$ are obtained, including Griesmer codes, near Griesmer codes and distance-optimal codes.

翻译：本文通过定义集构造，从${\mathbb F}_{q}^m$的一般单纯复形出发，构造了一大类${\mathbb F}_{q}$上的射影线性码，推广了[IEEE Trans. Inf. Theory 66(11):6762-6773, 2020]中的结果。完全确定了这类码的参数和重量分布。利用Griesmer界，我们给出了码为Griesmer码的充要条件以及码为距离最优码的充分条件。对于一种特殊情况，我们还给出了码为近Griesmer码的充要条件。此外，通过分别讨论具有一个、两个和三个极大元的单纯复形情形，更明确地给出了码的参数和重量分布，表明码分别至多为$2$-重量、$5$-重量和$19$-重量。通过详细研究这三种情形下码的最优性，获得了${\mathbb F}_{q}$上许多具有少数重量的最优线性码的无穷族，包括Griesmer码、近Griesmer码和距离最优码。