On the construction of permutation arrays via mappings from binary vectors to permutations

Abstract

An (n, d, k)-mapping f is a mapping from binary vectors of length n to permutations of length n + k such that for all x, y is an element of {0, 1}(n), d(H) (f (x), f (y)) >= d(H)(x, y) + d, if d(H) (x, y) <= (n + k)-d and d(H) (f (x), f (y)) = n + k, if d(H) (x, y) > (n + k)-d. In this paper, we construct an (n, 3, 2)-mapping for any positive integer n >= 6. An (n, r)-permutation array is a permutation array of length n and any two permutations of which have Hamming distance at least r. Let P (n, r) denote the maximum size of an (n, r)-permutation array and A (n, r) denote the same setting for binary codes. Applying (n, 3, 2)-mappings to the design of permutation array, we can construct an efficient permutation array (easy to encode and decode) with better code rate than previous results [Chang (2005). IEEE Trans inf theory 51:359-365, Chang et al. (2003). IEEE Trans Inf Theory 49:1054-1059; Huang et al. (submitted)]. More precisely, we obtain that, for n >= 8, P(n, r) >= A(n-2, r-3) > A(n-1, r-2) = A (n, r-1) when n is even and P(n, r) >= A(n-2, r-3) = A(n-1, r-2) > A(n, r-1) when n is odd. This improves the best bound A(n-1, r-2) so far [Huang et al. (submitted)] for n >= 8.