Existence of GBRDs with block size 4 and BRDs with block size 5

Chaudhry et al. (J Stat Plann Inference 106:303–327, 2002) have examined the existence of BRD(v, 5, λ)s for ({lambda in {4, 10, 20}}). In addition, Ge et al. (J Combin Math Combin Comput 46:3–45, 2003) have investigated the existence of ({{rm GBRD}(v,4,lambda; mathbb{G}){rm s}}) when ({mathbb{G}}) is a direct product of cyclic groups of prime orders. For the first problem, necessary existence conditions are (i) v ≥ 5, (ii) λ(v ? 1) ≡ 0 (mod4), (iii) λ v(v ? 1) ≡ 0 (mod 40), (iv) λ ≡ 0 (mod 2). We show these are sufficient, except for ({v=5, lambda in {4,10}}). For the second problem, we improve the known existence results. Five necessary existence conditions are (i) v ≥ 4, (ii) ({lambda equiv 0;({rm mod},|mathbb{G}|)}), (iii) λ(v ? 1) ≡ 0 (mod 3), (iv) λ v(v ? 1) ≡ 0 (mod 4), (v) if v = 4 and ({|mathbb{G}| equiv 2;({rm mod},4)}) then λ ≡ 0 (mod 4). We show these conditions are sufficient, except for ({lambda = |mathbb{G}|, (v,|mathbb{G}|) in {(4,3), (10,2), (5,6), (7,4)}}) and possibly for ({lambda = |mathbb{G}|, (v,|mathbb{G}|) in {(10,2h), (5,6h), (7,4h)}}) with h ≡ 1 or 5 (mod 6), h > 1.