Spreads Of Lines and Regular Group Divisible Designs

Abstract

A $1$-spread of a BIBD $\mathcal{D}$ is a set of lines of maximal size of $\mathcal{D}$ which partitions the point set of $\mathcal{D}$. The existence of infinitely many non-symmetric BIBDs which (i) possess a $1$-spread, and (ii) are not merely a multiple of a symmetric BIBD,
is shown. It is also shown that a $1$-spread $\mathcal{S}$ gives rise to a regular group divisible design $\mathcal{G}(\mathcal{S})$. Necessary and sufficient conditions that the dual of such a group divisible design $\mathcal{G}(\mathcal{S})$ be a group divisible design are established and used to show the existence of an infinite class of symmetric regular group divisible designs whose duals are not group divisible.