# 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.