Largest Permutation Codes With the Kendall <inline-formula> <tex-math notation="LaTeX">$\tau $ </tex-math> </inline-formula>-Metric in <inline-formula> <tex-math notation="LaTeX">$S_{5}$ </tex-math> </inline-formula> and <inline-formula> <tex-math notation="LaTeX">$S_{6}$ </tex-math> </inline-formula>

Abstract

An important problem in the theory of permutation codes is finding the value of $P(n,d)$ , the size of the largest subset of the set of all permutations $S_{n}$ with minimum Kendall $\tau $ -distance $d$ . Using an integer programming approach, we find the values of $P(5,d)$ for $d \geq 3$ and $P(6,d)$ for $d \geq 4$ . We give instances of codes which achieve these values. We also show that $P(6,3) \geq 102$ by giving a code of cardinality 102 in $S_{6}$ , which has minimum Kendall $\tau $ -distance 3.