Computing Transverse $t$-Designs

Abstract

In this paper, we develop a computational method for constructing transverse $t$-designs. An algorithm is presented that computes the $G$-orbits of $k$-element subsets transverse to a partition $\mathcal{H}$, given that an automorphism group $G$ is provided. We then use this method to investigate transverse Steiner quadruple systems. We also develop recursive constructions for transverse Steiner quadruple systems, and we provide a table of existence results for these designs when the number of points $v\le 24$. Finally, some results on transverse $t$-designs with $t>3$ are also presented.