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 $S_7$.

翻译：为了在非挥发性内存存储中高效写写和存储数据, 提议了等级调制方案。 级别调制方案错误更正是通过考虑调制代码完成的。 在本文中, 我们考虑的是所有对美元元素( $S_ n$, 使用Kendall $\ tau$- 美元) 进行调制的一组调制的代码。 本文的主要目标是从这种代码的大小中得出新的界限。 为此, 我们还考虑的是完美代码、 直径完美代码以及肯德尔 $\ tau$- 度量的顶级防毒码大小, 这些结构本身具有相当大的兴趣。 我们证明, 在美元( $n, $> 4美元为正元或 4\leq nleq 10美元) 上, 没有完美的单机重校正代码。 我们在已知的单倍- 7 美元 代码中, 我们提出了更大的单倍值值 。