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, 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.
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Communicated by R.C. Mullin.
The work was supported in part by the National Science Council of Taiwan under contract NSC-93-2213-E-009-117
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Huang, YY., Tsai, SC. & Wu, HL. On the Construction of Permutation Arrays via Mappings from Binary Vectors to Permutations. Des Codes Crypt 40, 139–155 (2006). https://doi.org/10.1007/s10623-006-0003-9
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DOI: https://doi.org/10.1007/s10623-006-0003-9