Perfect Permutation Codes with the Kendall's $\tau$-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, Sn, using the Kendall’s τ -metric. We prove that there are no perfect single-error-correcting codes in Sn, where n > 4 is a prime or 4 ≤ n ≤ 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 τ -metric and consider the related codes and specifically we prove the existence of a perfect single-error-correcting code in S5. Finally, we examine the existence problem of diameter perfect codes in Sn and obtain a new upper bound on the size of a code in Sn with even minimum Kendall’s τ -distance.