New Upper Bounds for the Size of Permutation Codes via Linear Programming
An $(n,d)$-permutation code of size $s$ is a subset $C$ of $S_n$ with $s$ elements such that the Hamming distance $d_H$ between any two distinct elements of $C$ is at least equal to $d$. In this paper, we give new upper bounds for the maximal size $\mu(n,d)$ of an $(n,d)$-permutation code of degree $n$ with $11\le n\le 14$. In order to obtain these bounds, we use the structure of association scheme of the permutation group $S_n$ and the irreducible characters of $S_n$. The upper bounds for $\mu(n,d)$ are determined solving an optimization problem with linear inequalities.