Group Divisible Designs with Three Groups and Block Size 4

Group divisible designs are classical combinatorial designs studied for their applications as well as for their own sake. They provide an ample opportunity of developing techniques to study combinatorial design constructions. GDDs are inherently hard to construct, especially when the number of groups is less than the block size and group sizes are different. The subject matter for this chapter is GDDs of block size four with three groups of different sizes. A previous study of the problem addressed the cases when the first group size, say \(n_1\) is 1 or 2, the second group size \(n_2=n\) greater than or equal to \(n_1\) and the third group size is \(n + 1\). The first part of the present paper tackles again the case of the first group having size one and the third group having size \(n + 2\). We also obtain several non-existence results when restrictions on block configurations are placed. The second part of the chapter deals with group sizes 3, n \( (n \ge 3)\) and \(n + 1\), respectively. We hope that these constructions of specific families will help to develop a more unified approach to construct such GDDs.