The fundamental construction for 3-designs

We discuss a generalization of Wilson's fundamental construction for group divisible designs which is intended to produce 3-wise balanced designs, rather than pairwise balanced designs. The construction generalizes many known recursive constructions for Steiner quadruple systems and other related designs. The generalization of Wilson's construction is based on a structure which is a 3-design analogue of a group divisible design. We show that other such analogues which have appeared in the literature are special cases of our definition. We also give several new applications of these structures.