Semidefinite programming for permutation codes

We initiate study of the Terwilliger algebra and related semidefinite programming techniques for the conjugacy scheme of the symmetric group Sym(n). In particular, we compute orbits of ordered pairs on Sym(n) acted upon by conjugation and inversion, explore a block diagonalization of the associated algebra, and obtain improved upper bounds on the size M(n,D)M(n,D) of permutation codes of lengths n=6,7n=6,7. For instance, these techniques detect the nonexistence of the projective plane of order six via M(6,5)<30M(6,5)<30 and yield a new upper bound M(7,4)≤535M(7,4)≤535 for a challenging open case. Each of these represents an improvement on earlier Delsarte linear programming results.