A Generalization of Group Divisible Designs

Abstract

Roy [8] extended the idea of Group Divisible designs of Bose and Connor [1] to $m$-associate classes, calling such designs Hierarchical Group Divisible designs with $m$-associate classes. Subsequently, no literature is found in this direction. The purpose of this paper is to study these designs systematically. A compact definition of the design, under the name Group Divisible $m$-associate (GD $m$-associate) design is given in Section 2. In the same section the parameters of the design are obtained in a slightly different form than that of Roy. The uniqueness of the association scheme from the parameters is shown in Section 3. The designs are divided into $(m + 1)$ classes in Section 4. Some interesting combinatorial properties are obtained in Section 5. The necessary conditions for the existence of a class of these designs are obtained in Section 7. Finally, some numerical illustrations of these designs are given in the Appendix.