A Study Of The Large Set Of Latin Squares With Holes

The large set problem is an important research object in combinatorial designs.In this thesis,we investigate the existence of large sets of partitioned incomplete Latin squares(LSPILS).The first chapter is an introduction,where some historical backgrounds,definitions of the large set problems are recalled and our main methods and results are also listed.The second chapter focuses on the existence of large sets of symmetric partitioned incomplete Latin squares of type gn(LSSPILS),which can be viewed as a generalization of the well known golf designs and the existence of large sets of partitioned incomplete Latin squares of type gn(LSPILS).Constructions for LSSPILSs are presented from some other large sets,such as golf designs,large sets of group divisible designs and large sets of Room frames.We prove that there exists an LSSPILS(gn)if and only if n ? 3,g(n-1)? 0(mod 2),and(g,n)?(1,5).We complete the determination of the spectrum of an LSPILS(gn)by viewing a partitioned incomplete Latin square as a corresponding regular bipartite graph and applying the known results of one-factors in graph theory.Thus,we prove that there exists an LSPILS(gn)if and only if g?1,n?3,and(g,n)?(1,6).In the third chapter,we define the large set‘plus'of disjoint incomplete Latin squares(LDILS+),which has additional properties.And we prove that there exists an LDILS+(n+a,a)if and only if n?1,1<a?n,(n,a)?(2,1),(6,5).The fourth chapter discusses the existence of large sets of partitioned incomplete Latin squares with two group sizes.Algebraic methods and combinatorial methods are employed to construct the large sets of partitioned incomplete Latin squares,as well as applying algebraic constructions such as quasi-difference matrix,and the knowledge of finite field to obtain infinite classes of large sets of partitioned incomplete Latin squares.Additionally,we define some new combinatorial structures such as ordered partitionable candelabra systems(POCS)and ordered generalized frames(OF),combined with classic ones such as 3-BD designs,to present constructions of the large sets of partitioned incomplete Latin squares.Furthermore,we introduce the definition of the large set‘plus'of partitioned incomplete Latin squares,and present the innovative construction of adding new points in the weighting construction.For the prime power q?3,we basically solve the existence of LSPILS+(gqu1)with some possible exceptions.For the positive integer n and u=2,3,we obtain the infinite classes of the existence of LSPILS+(gn(gu)1).The fifth chapter presents constructions of LHMTS+(gnu1)and LGDD+(gnu1),and prove that(1)there exists an LHMTS+(g3m(2g)1)for g,m? 1;(2)there exists an LGDD(g2n(2gn-g)1)for g?1 and n?2;(3)there exists an LGDD(g3v+1(3g)1)for v=5k and g,k?1.We also present a construction of orthogonal LSPILS+(1p21)s,where p is a prime and p?1(mod 6).For u>2,we discuss the generalized construction of orthogonal LSPILS(1qu1)s.The last chapter proposes some unsettled problems in this thesis.