No abstract

We show the necessary conditions are su cient for the existence of GDD(n; 2; 4; 1, 2) with two groups and block size four in which every block intersects each group exactly twice (even GDD's) or in which every block intersects each group in one or three points (odd GDD's). We give a construction for near 3-resolvable triple systems TS(n; 3; 6) for every n ¿ 4, and these are used to provide constructions for several families of GDDs.

We give some constructions of new infinite families of group divisible designs, GDD(n, 2, 4; 1 , 2 ), including one which uses the existence of Bhaskar Rao designs. We show the necessary conditions are sufficient for 3 n 8. For n = 10 there is one missing critical design. If 1 > 2 , then the necessary conditions are sufficient for n ≡ 4, 5, 8 (mod 12). For each of n=10, 15, 16, 17, 18, 19, and 20 we indicate a small minimal set of critical designs which, if they exist, would allow construction of all possible designs for that n. The indices of each of these designs are also among those critical indices for every n in the same congruence class mod 12.

Several new families of c-Bhaskar Rao designs with block size 4 are constructed. The necessary conditions for the existence of a c-BRDðv v; 4; kÞ are that:It is proved that these conditions are necessary, and are sufficient for most pairs of c and k; in particular, they are sufficient whenever k À c 6 ¼ 2 for c ! 0 and whenever c À k min 6 ¼ 2 for c < 0. For c > 0, the necessary conditions are sufficient for v v > 101; for the classic Bhaskar Rao designs, i.e., c ¼ 0, we show the necessary conditions are sufficient with the possible exception of 0-BRDðv v; 4; 2Þ's for v v 4 ðmod 6Þ.