Cyclic Difference Families And Related Designs

Let K be a set of positive integers, g and v positive integers. A (gv, g, K,λ)-difference family in Zgv ((gv, g, K,λ)-DF in Zgv for short) is a family F of subsets (called base blocks) of Zgv that satisfy, (1) if B∈F, then |B|∈K, and (2) the dif-ference list△F= U_B∈F△B is exactlyλtimes of Zgv\{0,v,2v,…, (g-1)v}, where△B={x-y:x,y∈B,x≠y}.Difference family is one of the most important concepts in design theory, and it is the generalization of the concept of difference set. The method of difference families plays an important role in constructing other kinds of designs. The known existence results and construction methods of cyclic designs are limited. In this thesis, we try to investigate the existence of cyclic difference families, which has close relation to cyclic group divisible designs. We also do research on the existence problem of cyclic di-rected difference families. What is more, we discuss the existence of perfect difference families, which is a powerful tool to construct optimal optical orthogonal codes.This thesis is organized as follows.Chapter 1 gives a brief introduction on the background of cyclic difference family and the main result of this thesis.In Chapter 2, some auxiliary designs are introduced to establish the recursive con-structions for strictly cyclic difference families. Both direct and recursive construc-tions are used to show the necessary and sufficient conditions for the existence of a (gv, g,3,λ)-SDF in Zgv. Then the existence spectrum of a cyclic (3,λ)-GDD of type gv having no short orbits is determined.In Chapter 3, the direct construction of finding difference triples, and also the recur-sive constructions and known results of strictly cyclic difference families from Chapter 2 are used, to show that the necessary conditions for the existence of a (gv,{g,3α},3,α)-SDF in Zgv forα= 1,3 are also sufficient, with the exception of (g, v)= (1,9) whenα=1. The existence spectrum of a cyclic (3,α)-GDD of type gv having a short orbits forα=1,3 is then established.In Chapter 4, applying the recursive constructions from Chapter 2, and the known results of strictly cyclic difference families from Chapter 2 and Chapter 3, the necessary and sufficient conditions for the existence of a (gv,{g,3α},3,λ)-SDF in Zgv forα≤ A is obtained. Meanwhile, it is shown that a cyclic (3,λ)-GDD of type gv havingαshort orbits exists if and only if (1)λg(v-1)-2α≡0 (mod 6),α≤λ, v≥3; (2) v(?)2,3 (mod 4) when g≡2 (mod 4) andλ≡1 (mod 2); (3) v(?)2 (mod 4) when g≡1 (mod 2) andλ≡2 (mod 4); (4) g(?)0 (mod 3) and v≡0 (mod 3) whenα≠0; (5)λ(3g-1)-2αg≡0 (mod 6) when v≡3; (6)λ=αwhen (g, v)=(1,3),λ=2αwhen (g,v)= (2,3),λ=4αwhen (g,v)= (1,6),λ≥2αwhen (g,v)=(2,6),λ= 0 (mod 3) when (g, v)= (1,9) andλ=α.In fact, with the known results of strictly cyclic difference families, the existence of cyclic difference families with the same parameter is certainly obtained. It is proved that the necessary conditions for the existence of a (gv,{g,3α},3,λ)-DF in Zgv forα∈[0,2] andα≤λare also sufficient, with two exceptions of (v,g,λ,α)= (9,1,1,1), (9,1,2,2). Finally, the existence spectrum of a cyclic (3,λ)-GDD of type gv is determined.In Chapter 5, the concept of perfect cyclic directed difference family is introduced, and some recursive constructions for cyclic directed difference families are established. Several finite families for the existence of (gv,g, k, 1)-dDF in Zgv when k= 4,5 are presented.In Chapter 6, it is shown that an ASP(2r+1,3) exists for 2≤r≤100 and r≠4,5. The existence of a (v,4,1)-PDF is investigated by taking advantage of the relationship between ASPs and perfect difference families (PDFs). It is proved that a (12t+1,4,1)-PDF exists for t≤100 and t≠2,3. Several recursive constructions for ASPs and PDFs are also presented. As a consequence, the existence results of an optimal (v,4,1)-OOC are updated.