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

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