Spreads Of Lines and Regular Group Divisible Designs

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.