Distance-increasing mappings from binary vectors to permutations that increase hamming distances by at least two

In this correspondence, for any k ges 2, we first propose two constructions of (n,k) distance-increasing mappings (DIMs) from the set of binary vectors of length n to the set of permutations of the same length that strictly increase the Hamming distance by at least k except when it is obviously not possible. Next, we prove that for any k ges 2, there is a smallest positive integer n_k such that an (n,k) DIM can be constructed for any n ges n_k. An explicit upper bound on n_k is also given