Largest Permutation Codes With the Kendall $\tau $ -Metric in $S_{5}$ and $S_{6}$

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 Sn with minimum Kendall τ-distance d. Using an integer programming approach, we find the values of P(5, d) for d ≥ 3 and P(6, d) for d ≥ 4. We give instances of codes which achieve these values. We also show that P(6, 3) ≥ 102 by giving a code of cardinality 102 in S₆, which has minimum Kendall τ-distance 3.