Constructions Of Doubly Resolvable Designs

Doubly resolvable Steiner systems,including the large set of Kirkman triple systems proposed by Sylvester in 1861,are classical combinatorial structures in design theory.Their constructions and existence have attracted much attention.However,the research progress is quite slow and the known results are very limited.In this thesis,we mainly study constructions of doubly resolvable Steiner quadruple systems(DRSQSs),large sets of Kirkman triple systems(LKTSs),and some related combinatorial structures.By introducing doubly resolvable Steiner quadruple system with intersecting property(DRSQS*),new recursive constructions for DRSQSs and LKTSs are established,and some new existence results are obtained.This thesis is organized as follows.Chapter 1 describes backgrounds and our main results.Chapter 2 focuses on constructions of DRSQSs.We first introduce the combinatorial structure(2,1)(1,1)-DRSQS*(v),then use it to construct a(2,1)(1,1)-DRSQS(4v)as well as a(2,3)(1,1)-DRSQS(8v).Also,product constructions and a recursive construction based on 3-wise balanced designs for DRSQS*s are established.Finally,we show that there is a(2,1)(1,1)-DRSQS*(4n)and a(2,3)(1,1)-DRSQS(22n+1)for any positive integer n.Chapter 3 pays attention to OLKFs and LKFs,which are auxiliary designs in the recursive constructions of LKTSs.By introducing a concept of δ-FPHF,we establish some recursive constructions for OLKFs and LKFs based on 3-wise balanced designs.We also give constructions for OLKFs via frames and constructions for LKFs via(2,1)-resolvable designs,and accordingly obtain some new existence results for OLKFs and LKFs.Chapter 4 is concerned with constructions for LKTSs.We utilize a(2,1)-RSQS*(v),which is a weaken structure of DRSQS*,to construct an LKTS(2v+1)and obtain a new infinite class of LKTS(22n+1+1).We also use 3-wise balanced designs to present recursive constructions for LR-designs,which play an important role in the constructions of LKTSs.Also,some constructions for OLKTSs are established incidentally,and some new existence results of LKTSs,LR-designs and OLKTSs are given accordingly.Chapter 5 summaries this thesis and poses some problems for future work.