On relative $t$-designs in polynomial association schemes

TL;DR

This paper explores two types of relative t-designs within polynomial association schemes, utilizing Terwilliger algebra to derive bounds and characterizations, especially focusing on Hamming schemes and their algebraic properties.

Contribution

It introduces a new algebraic framework for relative t-designs in association schemes and characterizes Hamming schemes through these designs and algebraic methods.

Findings

Explicit Fisher type lower bounds on the sizes of relative t-designs.

Equivalence of the two relative t-designs in Hamming schemes.

A new algebraic characterization of Hamming 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.