On the Construction of Permutation Arrays via Mappings from Binary Vectors to Permutations

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 ∈ {0, 1}n , dH ( f (x), f (y)) ≥ dH (x, y) + d , if dH (x, y) ≤ (n+k)−d and dH ( f (x), f (y)) = n+k, if dH (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.