Infinite Families of Optimal Linear Codes Constructed from Simplicial Complexes

Abstract

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 CΔc constructed from simplicial complexes in Fn2, where Δ is a simplicial complex in Fn2 and Δc the complement of Δ. We first find an explicit computable criterion for CΔ c to be optimal; this criterion is given in terms of the 2-adic valuation of Σi=1 s 2|Ai|-1, where the Ai '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 CΔc is 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. © 1963-2012 IEEE.