Large Sets Of Idempotent Latin Squares With Restricted Conditions

Large sets of designs and group divisible designs are important research topics in design theory.Here is an introduction on large sets of designs.Kirkman posed a problem of15 schoolgirls in 1850:Fifteen young ladies in a school walk out three abreast for seven days in succession:it is required to arrange them daily,so that no two shall walk twice abreast.In modern language,we are to find out a KTS(15).An interesting extension was suggested by Sylvester:Can the 455 triples of the elements of a 15-set be grouped into 13 different KTS(15)s?This is the first large set problem in the mathematics history,i.e.,large sets of Kirkman triple systems(LKTS).Then various large sets problems are proposed and solved completely.The spectrum for large sets of resolvable idempotent Latin squares(LRILS)is completely solved with 13 possible exceptions.In 2015,professor Zhu proposed some large sets problems for idempotent Latin squares with restricted conditions,inwhich contain r-golf design(denoted by r-G(v))and r-large sets of Mendelsohn triple systems(denoted by r-LMTS(v)).Here is an introduction on group divisible designs.Group divisible 2-designs were investigated very extensively.So we concern on group divisible 3-designs with block size four.A uniform group divisible 3-design is called an H-design whose spectrum was determined,therefore the research on nonuniform group divisible 3-designs are much attentioned.Group divisible 3-designs with block size four and group type 1~ns~1are closely related to the Steiner quadruple systems and large sets of Steiner triple systems,so the existence of this kind of design is our concern.In this thesis,we mainly discuss r-golf designs(r-G(v)),r-large sets of Mendel-sohn triple systems(r-LMTS(v)),large sets of resolvable idempotent Latin squares(LRILS(v))and group divisible 3-designs with block size four and group type 1~ns~1.This thesis is divided into six chapters.In chapter 1,we introduce some definitions,research backgrounds about large sets of designs,group divisible designs and give the main results of this thesis.In chapter 2,the existence problem of an r-G(v)is researched,which can be con-sidered as the generation of golf designs(G(v))and self-converse large sets of pure Mendelsohn triple systems(LPMTS~*(v)).In this chapter,we give two tripling con-structions.Combining with the small examples obtained by direct construction,the existence results of some infinite classes of r-G(v)s are given.Finally,the concept of partitionable ordered candelabra systems(denoted by(a,b)-POCS)is given,which offers much flexibility in the parameter r of the r-G(v)s for odd v.We establish the existence of an infinite family of r-G(v)s with all admissible integers r which relatively to this class of v.In chapter 3,the existence problem of an r-LMTS(v)is researched,which is a generation of large sets of Mendelsohn triple systems.In this chapter,we first give the existence results of r-LMTS(q)s when r=0,1,2 and q is an odd prime power by direct-ed construction of algebra.At the same time,we give the existence of an r-LMTS(v)for all admissible(v,r)(which satisfy the necessary conditions)where v?12.Further-more,two tripling constructions and recursive constructions by partitionable ordered candelabra systems(denoted by(a,b)-POCS*)are given.Combining with direct and recursive constructions,some infinite classes existence results of r-LMTS(v)s are given.In chapter 4,the existence problem of large sets of resolvable idempotent Latin squares(LRILS(v))is researched.In this chapter,we first give the direct construction method of LRILS of order q+2 and q+1 and then give the existence of LRILS(v)s,v?{14,20,28,34,38,42,50,55,62}.Further,the existence of LRILS(v)s,v?{22,35,40,46}are given by directed construction using multiplier automorphism and symmetric group.Finally,the spectrum for LRILS(v)is completely determined.In chapter 5,the existence problem of group divisible 3-designs with block size4 and type 1~ns~1is researched.In this chapter,we give two methods to construct this design by candelabra quadruple systems,which makes full use of the existence results of candelabra quadruple systems with group sizes odd and even.At the same time,by using automorphism group and computer search,the existence results of some small examples are given.So some infinite classes are obtained by combining small exam-ples and recursive constructions.Finally,by using the basic construction of Hartman's candelabra systems and the existence results of GDD(3,4,n+s)s of type 1~ns~1,we get some existence results of candelabra quadruple systems with group sizes odd.In chapter 6,we give the summary of this paper and the problems to be further studied.