Group divisible designs with block size 4 and group sizes 2 and 5

Abstract

In this paper we provide a 4‐GDD of type 2 2 5 5 ${2}^{2}{5}^{5}$, thereby solving the existence question for the last remaining feasible type for a 4‐GDD with no more than 30 points. We then show that 4‐GDDs of type 2 t 5 s ${2}^{t}{5}^{s}$ exist for all but a finite specified set of feasible pairs ( t , s ) $(t,s)$.

Group divisible designs with block size 4 and group sizes 4 and 7

“…The existence of $$4$$‐GDDs with no more than $$30$$ points has been well investigated. In [18], the existence question was answered for all but three types, and the existence results for those three types were completed in [2, 4].…”

“…Other work on the existence of $$4$$‐GDDs has concentrated on whether the group sizes are congruent to $$0$$, $$1$$ or $$2\,(\mathrm{mod}\,3)$$ and on GDDs whose groups are of only two or three different sizes. See, for example, [1–7, 9, 10, 23].…”

“…In the past, several GDDs have been found directly by assuming the existence of a cyclic automorphism group, $${\mathbb{G}}$$. See for instance [1–6, 8–11, 22, 23]. In these designs, one assigns a selection of base blocks and then develops these base blocks to produce the blocks of the required design.…”

“…Theorem 2.1 (Abel et al [2,4], and Kreher and Stinson [18]). The feasible group types for a 4-GDD on at most 30 points are 1 4 , 2 4 , 3 4 , 1 Existence results have been extended to point sets with up to v = 50 points.…”

“…The remaining five blocks are p q r { , , , } 20 8 14 ∞ , p q q r { , , , } 19 9 20 16 , p p q r { , , , } 8 9 13 10 , p r r r { , , , } 10 9 15 17 and q q q r { , , , } 14 15 19 8 . Removing these 16 blocks from the 4-GDD of type 1 7 64 9 gives a 4-GDD of type (12,4) or (5,8); • v t s = 79 : ( , ) = (11, 5); (17,5), (10,9) or (3,13); • v t s = : ( , ) = (20, 5); • v t s = : ( , ) = (3, 16); • v t s = : ( , ) = (9, 13) or (2,17); • v t s = : ( , ) = (19, 9), (12,13) or (5,17).…”