Infinite Families of Optimal Linear Codes Constructed From Simplicial Complexes
Jong Yoon Hyun, Jungyun Lee, Yoonjin Lee · IEEE Transactions on Information Theory · 2020
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 CΔcconstructed from simplicial complexes in F2n, where Δ is a simplicial complex in F2nand Δcthe complement of Δ. We first find an explicit computable criterion for CΔcto be optimal; this criterion is given in terms of the 2-adic valuation of Σsj=12|Ai|-1, where the At's are maximal elements of Δ. Furthermore, we obtain much simpler criteria under various specific conditions on the maximal elements of Δ. In particular, we find that CΔcis a Griesmer code if and only if the maximal elements of Δ are pairwise disjoint and their sizes are all distinct. Specially, when f has exactly two maximal elements, we explicitly determine the weight distribution of CΔc.We present many optimal linear codes constructed by our method, and we emphasize that we obtain at least 32 new optimal linear codes.