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Semidefinite programming for permutation codes (1212.1185v3)
Published 5 Dec 2012 in math.CO, cs.IT, and math.IT
Abstract: 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 up to 7. For instance, these techniques detect the nonexistence of the projective plane of order six via $M(6,5)<30$ and yield a new best bound $M(7,4) \le 535$ for a challenging open case. Each of these represents an improvement on earlier Delsarte linear programming results.