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.