Nonexistence of perfect permutation codes under the Kendall $$\tau $$-metric

Abstract

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 \(t=2,3,4,5,~\text {or}~\frac{5}{8}\left( {\begin{array}{c}n\\ 2\end{array}}\right) < 2t+1\le \left( {\begin{array}{c}n\\ 2\end{array}}\right) \).