Nonexistence of perfect permutation codes under the Kendall τ-metric

11/03/2020

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by   Wang Xiang, et al.

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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 τ-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 τ-metric, for more values of n. Specifically, we present the recursive formulas for the size of a ball with radius r in S_n under the Kendall τ-metric. Further, We prove that there are no perfect t-error-correcting codes in S_n under the Kendall τ-metric for some n and t=2,3,4,or 5.