Constructions Of (m;5) And (m;6)-Splitting Systems

suppose m and t are integers such that 0 < t≤m, An (m,t)-splittingsystem is a pair (X,B) that satisfies for every Y ? X with |Y | = t, thereis a subset B of X in B,such that |B∩Y | = 2t or |(X\B)∩Y | = 2t .suppose m , t1 and t2 are integers such that t1+t2≤m, An (m,t1,t2)-separating system is a pair (X,B) which satisfies : for every P ? X, Q ?X with |P| = t1, |Q| = t2 and P∩Q =φ, there exists a block B∈Bfor which either P ? B, Q∩B =φor Q ? B, P∩B =φ. A SplittingSystem or Separating System is called uniform if every block has thesame cardinality m2 .In this paper, we first discuss some basic properties of the separatingsystem, and by means of it, we present many methods for constructionsof splitting system, including direct construction,recursive and productconstruction. In the direct construction, we generalize the CoppersmithTheorem in [1]. In the indirect constructions, we present many methodsto construct the splitting system. At last, we use a probabilistic methodto produce a su?cient condition for the existence of splitting system ;then give an upper bound of splitting system and at the same time,we point out that this upper bound is better than the one given in [1]Theorem 4.1 when m is far greater than t . In this paper, we mainlyconcentrate on the case t = 5 and t = 6.