A generalization of group divisible t $t$ -designs

Abstract

Cameron defined the concept of generalized $t$-designs, which generalized $t$-designs, resolvable designs and orthogonal arrays. This paper introduces a new class of combinatorial designs which simultaneously provide a generalization of both generalized $t$-designs and group divisible $t$-designs. In certain cases, we derive necessary conditions for the existence of generalized group divisible $t$-designs, and then point out close connections with various well-known classes of designs, including mixed orthogonal arrays, factorizations of the complete multipartite graphs, large sets of group divisible designs, and group divisible designs with (orthogonal) resolvability. Moreover, we investigate constructions for generalized group divisible $t$-designs and almost completely determine their existence for $t=2,3$ and small block sizes.