| |
| Partially balanced incomplete block designs (PBIBDs) have a long history and been extensively used in agriculture and industrial experiments. Since the book of Clatworthy on two-associate-class partially balanced designs was published in 1973, not much progress was made on the construction of these designs. Group divisible designs (GDDs) are an important type of PBIBDs with two associate classes. The existence of a GDD with block size k = 3 was completely settled by Fu, Rodger and Sarvate. In their works, the most difficult one was the case when the number of groups, m, is less than the block size k. The existence of GDDs when m < k is, in general, a difficult case to solve. Indeed, when k = 4, very little is known about the existence of such GDDs. In this paper, we present two general construction methods for GDDs. The first one is a generalization of Wilson's Fundamental Construction in combinatorial design theory. The second one is an extension of the traditional construction using double group divisible designs. As an application of our new construction methods, a complete solution is provided for GDDs that generalize all the eleven designs in the old table of Clatworthy which have block size four, three groups, and replication number at most 10. For these GDDs, no progress has been made until very recently. |
|
| Keywords:Combinatorics; Associate class; Double group divisible design; Group divisible design; Partially balanced incomplete block design |
|
