New upper bounds on the size of permutation codes under Kendall τ -metric

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 S_n of all permutations on {1,… ,n} with the minimum distance (at least) d under the Kendall τ -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 . Applying these methods, we reduce the known upper bound (n-1)!-1 for P ( n , 3) to (n-1)!-⌈n/3⌉ +2≤ (n-1)!-2 , whenever n≥ 11 is prime. 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.