On relative $t$-designs in polynomial association schemes

Abstract

Motivated by the similarities between the theory of spherical $t$-designs and that of $t$-designs in $Q$-polynomial association schemes, we study two versions of relative $t$-designs, the counterparts of Euclidean $t$-designs for $P$- and/or $Q$-polynomial association schemes. We develop the theory based on the Terwilliger algebra, which is a noncommutative associative semisimple $\mathbb{C}$-algebra associated with each vertex of an association scheme. We compute explicitly the Fisher type lower bounds on the sizes of relative $t$-designs, assuming that certain irreducible modules behave nicely. The two versions of relative $t$-designs turn out to be equivalent in the case of the Hamming schemes. From this point of view, we establish a new algebraic characterization of the Hamming schemes.

17 pages

References in corpus (7)

• Distance-regular graphs
• Commutative association schemes
• Some algebra related to $P$-and $Q$-polynomial association schemes
• New proofs of the Assmus-Mattson theorem based on the Terwilliger algebra
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array
• Sharp tridiagonal pairs
• The structure of a tridiagonal pair

Cited by in corpus (4)

• Tight Cramér-Rao type bounds for multiparameter quantum metrology through conic programming
• Tight relative $t$-designs on two shells in hypercubes, and Hahn and Hermite polynomials
• On the existence of tight relative 2-designs on binary Hamming association schemes
• Relative t-designs in binary Hamming association scheme H(n,2)