We give two methods that are 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 Sn\documentclass[12pt]{minimal}
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\begin{document}$$S_n$$\end{document} of all permutations on {1,⋯,n}\documentclass[12pt]{minimal}
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\begin{document}$$\{1,\dots ,n\}$$\end{document} with the minimum distance (at least) d under the Kendall τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau$$\end{document}-metric. The first method is an integer programming problem obtained from the transitive actions of Sn\documentclass[12pt]{minimal}
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\begin{document}$$S_n$$\end{document}. The second method can be applied to refute the existence of perfect codes in Sn\documentclass[12pt]{minimal}
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\begin{document}$$S_n$$\end{document}. Applying these methods, we reduce the known upper bound (n-1)!-1\documentclass[12pt]{minimal}
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\begin{document}$$(n-1)!-1$$\end{document} for P(n, 3) to (n-1)!-⌈n3⌉+2≤(n-1)!-2\documentclass[12pt]{minimal}
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\begin{document}$$(n-1)!-\lceil \frac{n}{3}\rceil +2\le (n-1)!-2$$\end{document}, whenever n≥11\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 11$$\end{document} is prime. If n=6\documentclass[12pt]{minimal}
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\begin{document}$$n=6$$\end{document}, 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.
