Group‐divisible designs with block size k having k + 1 groups, for k = 4, 5

We investigate the spectrum for k-GDDs having k + 1 groups, where k = 4 or 5. We take advantage of new constructions introduced by R. S. Rees (Two new direct product-type constructions for resolvable group-divisible designs, J Combin Designs, 1 (1993), 15–26) to construct many new designs. For example, we show that a resolvable 4-GDD of type g5 exists if and only if g ≡ 0 mod 12 and that a resolvable 5-GDD of type g6 exists if and only if g ≡ 0 mod 20. We also show that a 4-GDD of type g4m1 exists (with m > 0) if and only if g ≡ m ≡ 0 mod 3 and 0   0) if and only if g ≡ m ≡ 0 mod 4 and 0 < m ≤ 4g/3, with 32 possible exceptions. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 363–386, 2000