Perfect Permutation Codes with the Kendall's $τ$-Metric

The rank modulation scheme has been proposed for efficient writing and storing data in non-volatile memory storage. Error-correction in the rank modulation scheme is done by considering permutation codes. In this paper we consider codes in the set of all permutations on $n$ elements, $S_n$, using the Kendall's $\tau$-metric. We prove that there are no perfect single-error-correcting codes in $S_n$, where $n>4$ is a prime or $4\leq n\leq 10$. We also prove that if such a code exists for $n$ which is not a prime then the code should have some uniform structure. We define some variations of the Kendall's $\tau$-metric and consider the related codes and specifically we prove the existence of a perfect single-error-correcting code in $S_5$. Finally, we examine the existence problem of diameter perfect codes in $S_n$ and obtain a new upper bound on the size of a code in $S_n$ with even minimum Kendall's $\tau$-distance.