Bounds on the Size of Permutation Codes with the Kendall $\tau$-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, $S_n$, using the Kendall $\tau$-metric. The main goal of this paper is to derive new bounds on the size of such codes. For this purpose we also consider perfect codes, diameter perfect codes, and the size of optimal anticodes in the Kendall $\tau$-metric, structures which have their own considerable interest. 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 present lower bounds on the size of optimal anticodes with odd diameter. As a consequence we obtain a new upper bound on the size of codes in $S_n$ with even minimum Kendall $\tau$-distance. We present larger single-error-correcting codes than the known ones in $S_5$ and $S_7$.