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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 [IEEETrans. Inf. Theory 66(11):6762-6773, 2020]. The parameters and weightdistribution of this class of codes are completely determined. By using theGriesmer bound, we give a necessary and sufficient condition such that thecodes are Griesmer codes and a sufficient condition such that the codes aredistance-optimal. For a special case, we also present a necessary andsufficient condition for the codes to be near Griesmer codes. Moreover, bydiscussing the cases of simplicial complexes with one, two and three maximalelements respectively, the parameters and weight distributions of the codes aregiven more explicitly, which shows that the codes are at most $2$-weight,$5$-weight and $19$-weight respectively. By studying the optimality of thecodes for the three cases in detail, many infinite families of optimal linearcodes with few weights over ${\mathbb F}_{q}$ are obtained, including Griesmercodes, near Griesmer codes and distance-optimal codes.

New constructions of optimal linear codes from simplicial complexes