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.

Plateaued functions are very important cryptographic functions due to their desirable cryptographic characteristics. We find explicit criteria for the construction of p-ary r-plateaued functions with an odd prime p. We point out that 0-plateaued functions are bent functions, and so plateaued functions generalize the notion of bent functions. We first derive an explicit form for the Walsh-Hadamard transform of a p-ary r-plateaued function. We then obtain an upper bound on the degree of p-ary r-plateaued functions, and we classify p-ary (n - 1)-plateaued functions in n variables. We also obtain explicit criteria for the existence of p-ary r-plateaued functions. Accordingly, these results lead to improved bounds on the existence of p-ary bent functions.