Nonexistence of perfect permutation codes under the l∞-metric

Permutation codes are studied due to their numerous applications in various applications, such as power line communications, block ciphers, and coding for storage. In this paper, we study perfect permutation codes in S-n, the set of all permutations on n elements, under the l(infinity)-metric. We present some nonexistence results on perfect t-error-correcting permutation codes in S-n, under the l(infinity)-metric for some t and n. More precisely, we prove that there does not exist a perfect t-error-correcting code in S, under the l(infinity)-metric for t and n satisfying 1 < t <= 3, 2t + 1 <= n or for t and n satisfying R2t+1 (n) = 0, 1, 2t, where 0 <= R-d (n) < d is the residue when dividing the positive integer n by the positive integer d.