Existence of 4-GDDs with at most 50 points and 4-GDDs of types 6 3 and 9 3

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

“…Theorem 4.3 summarises the direct constructions that are given in this paper; these are grouped according to their number of points, v. Proof. These designs are given in [2] except for type 2 5 2 8 which is given in [3]. □ Theorem 4.3.…”

“…For example, 4‐GDDs of types $${g}^{p}$$ and $${g}^{p}{n}^{1}$$ have been extensively studied; see for instance [5–7], [9], [10], [13] and [14]. Also, Abel et al [1-3] have determined the existence of 4‐GDDs of types $${3}^{t}{6}^{s}$$, $${3}^{t}{9}^{s}$$ and $${2}^{t}{8}^{s}$$ for all but a small finite number of specified cases.…”

“…Combined with $$v-{g}_{i}\equiv 0\,(\mathrm{mod}\,3)$$, this usefully simplifies to Condition 3 in Theorem 1.1. Conditions 7 and 8 in Theorem 1.1 come from [2].…”

“…, this usefully simplifies to Condition 3 in Theorem 1.1. Conditions 7 and 8 in Theorem 1.1 come from [2].…”

“…Lemma 3.1 (Brouwer et al [4]). There exists a 4-GDD of type g p if and only if (1) p 4 ≥ ; (2) ∕ . The existence of such designs has been determined in many cases; see for instance Forbes and Forbes [5], Forbes [6,7], Ge et al [10], Ge and Rees [9], Wei and Ge [14], and Rees [13].…”