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 λ∈{4, 10, 20} . In addition, Ge et al. (J Combin Math Combin Comput 46:3–45, 2003 ) have investigated the existence of GBRD(v,4,λ; 𝔾) s when 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, λ∈{4,10} . For the second problem, we improve the known existence results. Five necessary existence conditions are (i) v ≥ 4, (ii) λ≡ 0 ( mod |𝔾|) , (iii) λ( v − 1) ≡ 0 (mod 3), (iv) λ v ( v − 1) ≡ 0 (mod 4), (v) if v = 4 and |𝔾| ≡ 2 ( mod 4) then λ ≡ 0 (mod 4). We show these conditions are sufficient, except for λ = |𝔾|, (v,|𝔾|) ∈{(4,3), (10,2), (5,6), (7,4)} and possibly for λ = |𝔾|, (v,|𝔾|) ∈{(10,2h), (5,6h), (7,4h)} with h ≡ 1 or 5 (mod 6), h > 1.