New Upper Bounds on the Size of Permutation Codes under Kendall $\tau$-Metric

We first give two methods based on the representation theory of symmetric groups to study the largest size $P(n,d)$ of permutation codes of length $n$ i.e. subsets of the set $S_n$ all permutations on $\{1,\dots ,n\}$ with the minimum distance (at least) $d$ under the Kendall $\tau$-metric. The first method is an integer programming problem obtained from the transitive actions of $S_n$. The second method can be applied to refute the existence of perfect codes in $S_n$.\ Here we reduce the known upper bound $(n-1)!-1$ for $P(n,3)$ to $(n-1)!-\lceil \frac{n}{3}\rceil +2\le (n-1)!-2$, whenever $n\ge 11$ is any prime number. If $n=6$, $7$, $11$, $13$, $14$, $15$, $17$, the known upper bound for $P(n,3)$ is decreased by $3,3,9,11,1,1,4$, respectively.