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

  • Authors:

  • Yen-Ying Huang;Shi-Chun Tsai;Hsin-Lung Wu

  • Affiliations:

  • Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan 30050;Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan 30050;Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan 30050

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2006

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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)) 驴 dH (x, y) + d, if dH (x, y) 驴 (n + k) 驴 d and dH (f(x), f(y)) = n + k, if dH (x, y) (n + k) 驴 d. In this paper, we construct an (n,3,2)-mapping for any positive integer n 驴 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 驴 8, P(n, r) 驴 A(n 驴 2, r 驴 3) A(n 驴 1,r 驴 2) = A(n, r 驴 1) when n is even and P(n, r) 驴 A(n 驴 2, r 驴 3) = A(n 驴 1, r 驴 2) A(n, r 驴 1) when n is odd. This improves the best bound A(n 驴 1,r 驴 2) so far [Huang et al. (submitted)] for n 驴 8.