Some direct constructions of cyclic \((3, \lambda)\)-GDD of type \(g^v\) having prescribed number of short orbits (Q2875887)

A \((k,\lambda)\)-group divisible design (GDD) of type \(g^v\) is a set \(X\) of elements, a partition \({\mathcal G}\) of \(X\) into \(v\) groups each of size \(g\), and a collection \({\mathcal B}\) of \(k\)-subsets of \(X\) (blocks) so that two elements occur together in no block if they are in the same group, but appear together in exactly \(\lambda\) blocks when they are from different groups. It is cyclic when there is a cyclic group acting sharply transitively on \(X\) as an automorphism. Such an automorphism can partition the blocks into orbits of length \(|X|\) and into so-called short orbits. In [\textit{X. Wang} et al., Discrete Math. 311, No. 8--9, 663--675 (2011; Zbl 1229.05048)], a complete characterization of the existence of \((3,\lambda)\)-GDDs of type \(g^v\) is given, but certain GDDs needed in the constructions are left to the paper under review. These GDDs are explicitly constructed by providing the required difference families.