Semidefinite programming for permutation codes

TL;DR

This paper applies semidefinite programming and algebraic techniques to improve bounds on permutation codes in symmetric groups, achieving new upper bounds and nonexistence results for certain parameters.

Contribution

It introduces the use of the Terwilliger algebra and semidefinite programming for permutation codes, providing improved bounds and nonexistence proofs.

Findings

Improved upper bounds on permutation codes for n up to 7.

Detected nonexistence of the projective plane of order six.

Established a new bound M(7,4) ≤ 535.

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.