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In the rank modulation scheme for flash memories, permutation codes have beenstudied. In this paper, we study perfect permutation codes in $S_n$, the set ofall permutations on $n$ elements, under the Kendall \tau-Metric. We answer oneopen problem proposed by Buzaglo and Etzion. That is, proving the nonexistenceof perfect codes in $S_n$, under the Kendall \tau-metric, for more values of$n$. Specifically, we present the recursive formulas for the size of a ballwith radius $r$ in $S_n$ under the Kendall \tau-metric. Further, We prove thatthere are no perfect $t$-error-correcting codes in $S_n$ under the Kendall$\tau$-metric for some $n$ and $t$=2,3,4,or 5.

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