The Existence Of Mendelsohn Quadruple Systems

t-wise balanced designs are very important in design theory. For t = 2, much work has been done on pairwise balanced designs. For t = 3, however, not much is known. A 3-design with block size 4 is called a quadruple system ( QS(v, λ)). There are some ordered analogue of quadruple systems, such as Directed quadruple systems, Tetrahedral quadruple systems, Dihedral quadruple systems and Mendelsohn quadruple systems, etc.In this paper, we mainly investigate MQS(v,λ)s, MCQSλ(gn : s)s,MGDDλ(3, 4, ng)s of type gn. By direct and recursive constructions, we get the following results:(1) There is an MQS(v, λ) if and only if λv(v - 1) (v - 2)≡0 (mod 4) and v ≥ 4, except for v = 5, λ≡1 (mod 2).(2) There is an MCQSλ(g2 : 0) if and only if λ≥1, g≥2.(3) There is an MCQSλ(g3 : s) if and only if λg2(s - 1)≡0 (mod 2) and 0 ≤s≤g.(4) There is an MGDDλ(3, 4, ng) of type gn if and only if λn (n - 1) (n - 2)g3≡ 0 (mod 4) and n≥4, except for n = 5, λ≡1 (mod 2), g≡1 (mod 2).