Constructions Of (v,k,k-1)-DDFs In Z_v

Difference family is a kind of combinatorial design, and it is a generalization of difference set. Difference method is one of the most useful and effective for constructing combinatorial design of many types. For more information on difference family, the reader is referred to [13]. Let (G, +) be an Abelian group of order v and H a subgroup of G with g element. A (G, H, k, A) relative difference family (or (G, H, k, λ)-DF in short) is a collection T = {B_j, : i ∈ I} of k-subsets (called base blocks) of G with the property that its list of differences ΔF = ∪_i∈I ΔB_i is λ times G\H where ΔB_i = {a - b : a,b ∈ B_i,a ≠ b}. In the case that g = 1, we simply call it a (G, H, λ)-DF (or (v, k, λ)-DF over G).In this article, we mainly consider the following cases: The base block of {G, H, k, λ)-DF in G are pairwise disjoint. In particular G = Z_v, H — {0}, λ = k - 1, (G, H, k, λ)-DDF is denoted by (v, k,k- 1)-DDF in Z_v. We mainly study the sufficient condition of (u, k,k — 1)-DDF in Z_v, for k = 3,4. Disjoint difference families are related with many kinds of conbinatorial designs such as external difference families, whist tournament, cyclically almost resolvable cyclic directed triple systems and perfect bases.There are two chapters in the thesis:The first chapter introduces the basic definition, the relation between disjoint difference families and other designs, and some known results.In the second chapter, we will present some new constructions and recursive constructions of DDFs, in the last we get some new results.Firstly, we construct a (v, k — 1, k — 2)-DDF in Z_v through (v, k, 1)-CDF or (v,k,k,1)-CDF.Secondly, we use cyclic GDD and semi-cyclic frame to get a(v,g,k, k — 1)-DDF in Z_v.Finally, we give a special construction of (5p, 3, 2)-DDF in Z_5p, when p = 5(mod 12) is a prime.