Computing Transverse \(t\)-Designs

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 \( v \leq 24 \). Finally, some results on transverse \( t \)-designs with \( t > 3 \) are also presented.