  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 $(n-1)!-\lceil\frac{n}{3}\rceil+2\leq (n-1)!-2$, whenever $n\geq 11$ 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.
