A class of symmetric association schemes as inclusion of biplanes

Abstract

Let ${\cal B}$ be a nontrivial biplane of order $k-2$ represented by symmetric canonical incidence matrix with trace $1+ \binom{k}{2}$. We proved that ${\cal B}$ includes a partially balanced incomplete design with association scheme of three classes. Consequently, these structures are symmetric, having $2k-6$ points. While it is not known whether this class is finite or infinite, we show that there is a related superclass with infinitely many representatives.