On relative t-designs in polynomial association schemes

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 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.