2-designs having an intersection number $k-n$

In this paper we examine 2-designs having an intersection number k - n. This intersection number gives rise to an equivalence relation on the blocks of the design. Conditions on the sizes of these equivalence classes and some properties of any further intersection numbers are obtained. If such a design has at most three intersection numbers then it gives rise to a strongly regular graph. This leads to a result on the embedding of quasi-residual designs. As as example a quasi-residual 2-(56, 12, 3) design is constructed and embedded in a symmetric 2-(71, 15, 3) design.