New table of Bounds on Permutation Codes under Kendall τ-Metric

In order to overcome the challenges posed by flash memories, the rank modulation scheme was proposed. In the rank modulation the codewords are permutations. In this paper, we study permutation codes with a specified length and minimum Kendall $\tau$-distance, and with as many codewords (permutations) as possible. We managed to make many significant improvements in the size of the best known codes. In particular, we show that for all $n\geq 6$ and for all $\displaystyle \frac{3}{5}\begin{pmatrix}n\\2\end{pmatrix}\lt d\leq\frac{2}{3}\begin{pmatrix}n\\2\end{pmatrix}$ the largest size of a permutation code of length n and minimum distance at least d under Kendall $\tau$-metric is 4.