3-GDD(n1,n2,4; A1,A2)

We define a 3-GDD(ni,r»2>fc;Ai,A2) where ni n2 by extending the definitions of a group divisible design and a ir design. A 3-GDD(ni,n2, k; Ai, Aa) (where n r»2) is a set X of ni +7i2 elements partitioned into two parts of sizes nj and ri2, respectively, called groups together with a collection of fc-subsets of X called blocks, such that (i) every 3-subset of each group occur in At blocks and (ii) every 3-subset where two elements are from one group and one element from the other group occurs in A2 blocks. For the problem where ni = T12, it is generally written as a 3-GDD(n, 2, A:; Ai, A2), which was proposed and discussed in a previous study. In this paper, we study the necessary conditions for the existence of a 3-GDD(nx, n2, fc; Ai, A2) and present some families of such 3-GDDs as well as several examples these GDDs for small values of n and n2. We also show that the necessary conditions are sufficient for several special cases. © 2020 Charles Babbage Research Centre. All rights reserved.