Constructions Of Steiner Quadruple Systems With Resolvable Derived Designs And Their Related Designs

With the development of information technology,combinatorial design theory is intermingled with coding cryptography,network communication theory,computer science and information science,and various new combinatorial structures are introduced constantly.t-wise balanced design(t-BD)is an important combinatorial structure in combinatorial design theory.It has very important applications in coding theory and recursive constructions of various combinatorial structures.For example,3-BD can be used to construct large sets of Kirkman triple systems(LKTSs)and two-dimensional optical orthogonal codes(2-D OOCs)with good correlation characteristics.At the same time,the construction of t-BD is also closely related to finite field,group theory and other branches of mathematics.So t-BD has attracted much attention of many scholars.The existence problem of LKTSs proposed by Sylvester in 1861,which is the first large set problem in mathematics history and still far from settled.Steiner quadruple systems with resolvable derived designs(RDSQSs)play an important role in the recursive constructions of LKTSs.In this thesis,we introduce a special combinatorial structure RDSQS*(v)and use it to construct RDSQSs.As a consequence,some new infinite classes of LKTSs are given.In order to study the existence of RDSQS*s,we also introduce Kirkman triple systems with intersecting property(KTS*s)and research their existence problem.As a generalization of 3-BD,the spectrum for group divisible 3-designs H(u,v,4,3)s is completely determined by Mills and Ji.So we concern on group divisible packings HP(u,v,4,3)s with semi-cyclic property(semi-cyclic HP(u,v,4,3)s),which are closely related to 2-D(u×v,4,2)-OOCs with AM-OPPW restriction(AM-OPPW 2-D(u×v,4,2)-OOCs).Because 2-D OOCs enable optical communication at a lower chip rate,the research on semi-cyclic HP(u,v,4,3)s is also of great practical interest.In this thesis,we mainly discuss the constructions of RDSQSs,KTS*s,and semicyclic HP(u,v,4,3)s.This thesis is divided into four chapters.In Chapter 1,we study the constructions of RDSQS(v)s.We introduce a special combinatorial structure RDSQS*(v),and use it to present a construction for RDSQS(4v).We also use group divisible designs with resolvable derived designs(RDGDDs)to construct new RDSQSs.Applying the known product construction for RDSQSs,we obtain some new infinite classes of RDSQSs.Consequently,the existence results on LKTSs are improved.In Chapter 2,we study the existence problem of a KTS*(v),which is the derived design of an RDSQS*(v+1)at each point.We first give the existence results of some small orders by computer search.Then we give the direct construction methods of KTS*(3p)s when prime p≡1,5(mod 6).Combining with direct and recursive constructions,some infinite classes of KTS*(v)s are obtained.In Chapter 3,we study the constructions of semi-cyclic HP(u,v,4,3)s and their applications in 2-D OOCs.To give the basic recursive methods of semi-cyclic HP(u,v,4,3)s,we first introduce some auxiliary designs,for example,semi-cyclic IH-designs,semi-cyclic HGDDs,etc.Based on these recursive methods,we give the constructions of auxiliary designs.At the same time,some key orders involved in recursive constructions are given by using automorphism groups and computer search.Combined with the given recursive constructions,some infinite classes of semi-cyclic HP(u,v,4,3)s are obtained.Finally,some new results of optimal AM-OPPW 2-D(u×v,4,2)-OOCs are obtained from the equivalence relation between semi-cyclic HP(u,v,4,3)s and AM-OPPW 2-D(u×v,4,2)OOCs.In Chapter 4,we summarize the content of this thesis and pose some questions for further research.