GDD(n,2,4;λ1,λ2)(n,2,4;λ1,λ2) with equal number of even and odd blocks

We prove that the necessary condition, n≡0n≡0(mod 3), is sufficient for the existence of GDD(n,2,4;3,4n,2,4;3,4) except possibly for n=18n=18. We prove that necessary conditions for the existence of group divisible designs GDD(n,2,4;λ1,λ2n,2,4;λ1,λ2) with equal number of even and odd blocks are sufficient for GDD(n,2,4;5n,7(n−1))(n,2,4;5n,7(n−1)) for all n≥2n≥2, GDD(7s,2,4;5s,7s−1)(7s,2,4;5s,7s−1) for all ss, GDD(5t+1,2,4;5t+1,7t)(5t+1,2,4;5t+1,7t) for t≡t≡ 0(mod 2) and GDD(5t+1,2,4;2(5t+1),14t)(5t+1,2,4;2(5t+1),14t) for all tt. To complete the existence of such GDDs, one needs to construct two more families: GDD(5t+1,2,4;5t+1,7t)(5t+1,2,4;5t+1,7t) for all odd tt, and GDD(35s+21,2,4;5s+3,7s+4)(35s+21,2,4;5s+3,7s+4) for all positive integers ss.