GBRDs with block size 3 over odd order groups and groups of orders divisible by 2 but not 4

Well-known necessary conditions for the existence of a generalized Bhaskar Rao design, GBRD(v, 3, λ;G) with v ≥ 4 are: (i) λ ≡ 0 (mod |G|), (ii) λ(v − 1) ≡ 0 (mod 2), (iii) λv(v − 1) ≡ 0 (mod 3), (iv ) if |G| ≡ 0 (mod 2) then λv(v − 1) ≡ 0 (mod 8). In this paper we show that these conditions are sufficient whenever (i) the group G has odd order or (ii) the order is of the form 2q for q = 3 or q an odd number which is not a multiple of 3.