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The rank modulation scheme has been proposed for efficient writing andstoring data in non-volatile memory storage. Error-correction in the rankmodulation scheme is done by considering permutation codes. In this paper weconsider codes in the set of all permutations on $n$ elements, $S_n$, using theKendall $\tau$-metric. The main goal of this paper is to derive new bounds onthe 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 provethat 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 optimalanticodes with odd diameter. As a consequence we obtain a new upper bound onthe size of codes in $S_n$ with even minimum Kendall $\tau$-distance. Wepresent larger single-error-correcting codes than the known ones in $S_5$ and$S_7$.

Bounds on the Size of Permutation Codes with the Kendall $\tau$-Metric