“…Then there exists a 4-SCGDD of type m n whenever m ≡ 0 (mod 6) and n ≥ 5, except when (m, n) =(6,5), and possibly when n = 5, m ≡ ±6(mod 36) and m ≥ 30.5.2The case m ≡ 2, 4 (mod6) In view of Theorems 5.1 and 5.2, we only need to consider n ≡ 1 (mod 3) and n ≥ 7.Lemma 5.11There exists a 4-SCGDD of type 2 n for any n ≡ 1 (mod 3), n ≥ 7 and n = 10; and there does not exist any 4-SCGDD of type 210 .Proof The result for each stated value of n is given in Proposition 3.3.Lemma 5.12There exists a 4-SCGDD of type 4 n for any n ≡ 1 (mod 3) and n ≥ 10; and there does not exist any 4-SCGDD of type 4 7 .Proof A computer search shows that there is no 4-SCGDD of type 4 7 . For n ∈ {10,13,16, 19,22,25,28,31, 34, 37, 40, 43, 49, 52, 58, 61, 67, 73, 79, 94, 97, 103, 109}, the desired design can be found in Theorem 2.7. Now, we take u = 9 for n ∈ E = {46, 55, 64, 82, 100, 172}; u = 15 for n ∈ F = {76, 106, 166}; or u = 12 for n ∈ H = {169}.…”