3-designs And Some Applications

Let K be a set of positive integers. A t-wise balanced design is a pair (X,B), where X is a set of v points and B is a set of subsets of X (called blocks), each of cardinality from K, such that every t-subset of X is contained in a unique block.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. In this thesis, we concentrate on the subjects related to 3-wise balanced designs. For convenience, we always simply write 3-design for 3-wise balanced design.Since the survey of Lindner and Rosa in 1978, one of the interests in the area of 3-designs has focused on establishing the existence of Steiner quadruple systems with a prescribed automorphism group. However only limited progress has been made. In this thesis, we try to enlarge the results of strictly cyclic Steiner quadruple systems and rotational Steiner quadruple systems. As applications of 3-designs, we limit our discussion to two main areas: optical orthogonal codes and hypergraph decompositions.This thesis is organized as follows.Chapter 1 gives a brief introduction on the background of t-designs. In Chapter 2, we introduce some auxiliary designs to establish the fundamental construction for strictly cyclic 3-designs. For constructing strictly cyclic Steiner quadruple systems, some more recursive constructions for strictly cyclic 3-designs containing block-size 4 are presented. Using these constructions we have many infinite families of strictly cyclic Steiner quadruple systems.In Chapter 3, a direct construction for rotational Steiner quadruple systems of order p+1 having a non-trivial multiplier automorphism is given, where p≡13 (mod 24) is a prime. We also give two improved product constructions. By these constructions, the known existence results of rotational Steiner quadruple systems are extended.In Chapter 4, an application of our constructions in Chapter 2 is given to optimal optical orthogonal codes of length v with weight 4 and index 2 (denoted by (v, 4,2)-OOC). The study of optical orthogonal codes is motivated by their applications in a fiber-optical code-division multiple access (CDMA) channel, and they have been stud-ied extensively for the past two decades. Many infinite families of optimal (v, 4,2)-OOCs are obtained. We also notice that there does not exist an optimal (v, 4,2)-OOC for any v≡0 (mod 24). Thus we introduce the concept of strictly cyclic maximal pack- ing quadruple systems to deal with the case of v≡0 (mod 24) for (v, 4,2)-OOCs. By our recursive constructions, some infinite families are given on strictly cyclic maximal packing quadruple systems. In Chapters 2 and 4, as corollaries, many known con-structions for Steiner quadruple systems and optimal (v, 4,2)-OOCs are unified by our constructions.In Chapter 5, an application of 3-designs to hypergraph decompositions is pre-sented. We show that a decomposition of a 3-uniform hypergraph K_v³ into a special kind of hypergraph K₄³- e exists if and only if v≡0,1,2 (mod 9) and v≥9. And it is established that a decomposition of a 3-uniform hypergraph K_v³ into a special kind of hypergraph K₄³ + e exists if and only if v≡0,1,2 (mod 5) and v≥7.