Optimal Permutation Codes and the Kendall's τ\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's $\tau$-metric. We will consider either optimal codes such as perfect codes or concepts related to optimal codes. 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 consider optimal anticodes and diameter perfect codes in $S_n$. As a consequence we obtain a new upper bound on the size of a code in $S_n$ with even minimum Kendall's $\tau$-distance. We define some variations of the Kendall's $\tau$-metric and consider the related codes. Specifically, we present perfect single-error-correcting codes in $S_5$ for these variations. Furthermore, using these variations we present larger codes than the known ones in $S_5$ and $S_7$ with the Kendall's $\tau$-metric. These codes have a large automorphism group.Comment: 18 page

Perfect Permutation Codes with the Kendall's τ\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'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.Comment: to be presented in ISIT201