The r-Hamming gap and distance-gap-preserving mappings from binary vectors to permutations

The well known Hamming distance between a pair of permutations (or strings) of the same length is simply the number of pairs of different digits in these permutations. It has been an interesting topic of research to find mappings from the binary vectors to permutations of the same length such that the Hamming distance is preserved or increased. As a natural variation we introduce the Hamming gap of radius r between two permutations, which is, in a way, equivalent to the number of digits where the corresponding pair of entries differ by at least r. This is called the r-Hamming gap. We first discuss the properties of this new concept. We then show mappings from binary vectors to permutations such that the images of a pair of binary vectors (at Hamming distance d) have r-Hamming gap at least d. We also show the generalization of our findings to permutations on Z_n (where n ≡ 0) instead of [n].