The Existence Of Disjoint Difference Family

Block design is an important part in combinatorial design theory. The research of block design is mainly focused on some block designs, such as balanced incomplete block design,resolvable balanced incomplete block design,symmetric balanced incomplete block design,group divisible design,transversal design,difference family ... and so on.Supposeνandλare positive integers,Κis a set of positive integers. If a block design(X, A) satisfying the following conditions:(1) |X|＝ν(2) {|A| |A∈A} (?) K, the elements of A are called blocks;(3) Each 2-subset of X is contained in exactlyλblocks, then we call (X, A) a pairwise balanced design ((ν,Κ,λ)-PBD in short). A(ν,Κ,λ)-PBDβis said to be balanced incomplete block design ((ν,Κ,λ)-BIBD in short) if the size of each block B∈βis k. The necessary conditions for the existence of a (ν,Κ,λ)-BIBD areλ(ν-1)≡0 (mod k-1) andλν(ν-1)≡0 (mod k(k-1)). Difference family can be used to construct many block designs. We present the definition of difference family as follows.Suppose G is a abelian group of orderνwith operation "＋" ,Κis a set of positive integers and S＝{B_i: 1≤i≤s}, where B_i＝{b_i1, b_i2, ..., b_iki}. If(1) Each nonzero element a∈G occurs in exactlyλtimes among the following differences:a＝b_ij-b_il, 1≤i≤s, 1≤j, l≤k_i, j≠l;(2) k_i∈K, 1≤i≤s, then we call S a (ν,Κ,λ)-DF. When G＝Z_v is a cyclic group of order v, it is denoted by (ν,Κ,λ)-CDF. When K has only one element k, it is denoted by (ν,Κ,λ)-DF. The necessary condition for the existence of a (ν,Κ,λ)-DF isλ(ν-1)≡0 (mod k(k-1)). If all the elements of B are mutually disjoint, then S is called a (ν,Κ,λ)-DDF. The necessary conditions for the existence of a (ν,Κ,λ)-DDF areλ(ν-1)≡0 (mod k(k-1)) andλ≤k-1. The existence of (ν,Κ, 1)-DF had been studied extensively when k＝3, 4, 5, 6 and 7([1], [2], [3], [4], [5], [6]). (ν,Κ,λ)-DDF is a special kind of (ν,Κ,λ)-DF. Little is known for the existence of (ν,Κ,λ)-DDFs. The existence of (ν,Κ,λ)-DDFs had been completed solved when k＝3 andλ＝1([7],[8]). When k＝3,λ＝2, v≡1 (mod 3) is a prime power, the existence of (ν,Κ,λ)-DDFs had been solved([9]). There are also some results when k＝4([10]).Weil's theorem([11]) on character sum estimates is very useful in the construction of combinatorial designs. For example, it can be used to construct difference families([3],[4],[5],[6]), V(m, t)s ([12],[13],[14]), APAVs([15]), optimal optical orthogonal codes([16]), generalized Steiner systems ([17]). In this paper, by using Weil's theorem on character sum estimates, an explicit lower bound for the existence of a (ν,Κ, 1)-DDF is obtained, where v＝k(k-1)t＋1 is a prime power. Applying this result with k＝4 and a computer search, it is proved that there exists a (ν, 4, 1)-DDF for each prime powerν≡1 (mod 12) and v≥13. For k＝4,λ＝2, it is also proved that there exists a (ν, 4, 2)-DDF for each prime power v≡1 (mod 6) and v≥7. The following are main results of this thesis.Theorem 1.2.1 If v≡1 (mod k(k-1)) is a prime power and v≥E², then there exists a(ν,Κ, 1)-DDF, whereTheorem 1.2.2 There exists a (ν, 4, 1)-DDF for each prime power v≡1 (mod 12) and v≥13.Theorem 1.2.3 There exists a (ν, 4, 2)-DDF for each prime power v≡1 (mod 6) and v≥7.In the last part, some problems for further research are presented.