Nonexistence of perfect permutation codes under the Kendall τ-metric

In the rank modulation scheme for flash memories, permutation codes have been studied. In this paper, we study perfect permutation codes in S-n, the set of all permutations on n elements, under the Kendall tau-metric. We answer one open problem proposed by Buzaglo and Etzion. That is, proving the nonexistence of perfect codes in S-n, under the Kendall tau-metric, for more values of n. Specifically, we present the polynomial representation of the size of a ball in S-n under the Kendall tau-metric for some radius r, and obtain some sufficient conditions of the nonexistence of perfect permutation codes. Further, we prove that there does not exist a perfect t-error-correcting code in S-n under the Kendall tau-metric for some n and t = 2, 3, 4, 5, or 5/8(n 2) < 2t + 1 <= (n 2).