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 ) of permutation codes of lengths n = 6 , 7 . For instance, these techniques detect the nonexistence of the projective plane of order six via M ( 6 , 5 ) < 30 and yield a new upper bound M ( 7 , 4 ) ≤ 535 for a challenging open case. Each of these represents an improvement on earlier Delsarte linear programming results.