Infinite families of optimal linear codes constructed from simplicial complexes

A linear code is optimal if it has the highest minimum distance of any linear code with a given length and dimension. We construct infinite families of optimal binary linear codes $C_{\Delta ^{c}}$ constructed from simplicial complexes in $\mathbb {F}^{n}_{2}$ , where $\Delta $ is a simplicial complex in $\mathbb {F}^{n}_{2}$ and $\Delta ^{c}$ the complement of $\Delta $ . We first find an explicit computable criterion for $C_{\Delta ^{c}}$ to be optimal; this criterion is given in terms of the 2-adic valuation of $\sum _{i=1}^{s} 2^{|A_{i}|-1}$ , where the $A_{i}$ ’s are maximal elements of $\Delta $ . Furthermore, we obtain much simpler criteria under various specific conditions on the maximal elements of $\Delta $ . In particular, we find that $C_{\Delta ^{c}}$ is a Griesmer code if and only if the maximal elements of $\Delta $ are pairwise disjoint and their sizes are all distinct. Specially, when $\mathcal {F}$ has exactly two maximal elements, we explicitly determine the weight distribution of $C_{\Delta ^{c}}$ . We present many optimal linear codes constructed by our method, and we emphasize that we obtain at least 32 new optimal linear codes.