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 $$\in$$ {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.