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 F-2(n), where. is a simplicial complex in F-n(2) and Delta(c) the complement of.. 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 s i=1 2| Ai|-1, where the Ai'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(.) is a Griesmer code if and only if the maximal elements of Delta are pairwise disjoint and their sizes are all distinct. Specially, when 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.