组合设计的大集及超大集

【作者】 田子红；

【导师】 康庆德；

【作者基本信息】 河北师范大学 ， 基础数学， 2003， 博士

【摘要】 一个区组设计是由两个有限集合X，B及它们之间的关联关系I组成的，记为D=（X，B，I），其中X为v元集，B为区组集，对于指定的设计D，若X上与D相应的全部构形可分拆为若干个B_i，使得每个（X，B_i，I）皆为一个与D同参数同类型设计，则称这些B_i构成一个v阶D-设计的大集。若v元集X是v+1元集Y的一个子集，而Y上与D相应的全部指定构形可分拆为若干个B_x，使得每个B_x恰是Y＼{x}上的一个与D同参数同类型的设计（其中x∈Y），则称这些B_x构成一个v阶D-设计的超大集。 本文中讨论了一些区组设计的大集及超大集。全文共分六章： 第一章 综述几种区组设计大集和超大集的定义、研究背景和当前的研究现状，并给出了一些具体例子。 第二章 讨论三类有向三元系MTS，DTS，HTS之间的一种关联关系：如果一个MTS（v，λ）的区组关联图是3-边可染色的，则可得到三个不相交的DTS（v，λ）及四个不相交的HTS（v，λ），且对应区组的基础集均一致。从而可由MTS（v，λ）某类型的大集或超大集得到同类型的DTS（v，λ）与HTS（v，λ）的大集和超大集。 第三章 研究纯的Directed三元系大集。给出了一系列的递归构造和结果，指出整个问题的解决可压缩为对LPDTS（p+2）（p为奇素数）的讨论。 第四章 对三类有向三元系超大集作了进一步研究，推广了已有的结论，给出了一些新的递归方法。 第五章 讨论幂等拟群的大集和超大集。特别给出了幂等对称拟群超大集的存在谱。 第六章 研究三类混合三元系的大集和超大集。基本上给出了LT_i（v，λ），OLT_i（v）（i=1，2）和LT₃（v，2λ），OLT₃（v，2）的存在谱。更多还原

【Abstract】 A block design is a triple D = (X, B,I), where X is a finite set (the point set) of order v, B is the block set and / is the incidence relation between X and B. For the given block design D, if all the blocks from X, which have the same type of the blocks in B, can be partitioned into some Bi, such that each (X, Bi, I) is a block design with the same parameters and the same type as D, then {(X, Bi,I)}i is called as large set of D-design of order v. On the other side, let |X| = v,|Y| = v + 1, and X Y, if all the blocks of the same type from Y can be partitioned into some Bx, such that each (Y\{x}, Bx, I} is a block design of order v for any element x ∈ Y with the same parameters and the same type as D, then {(Y\{x},Bx,I)}x is called an overlarge set of D-design of order v.In this thesis, we discuss the existence problem about some types of large sets and overlarge sets.In Chapter 1, we introduce some definitions, research background and the present situations about some kinds of large sets and overlarge sets of block designs, and give some examples.In Chapter 2, a relationship between three types of oriented triple systems MTS, DTS, HTS is discussed, i.e., if the block incidence graph of an MTS(v, A) is 3 edge-colorable, then we can get three disjoint DTS(v, λ)s and four disjoint HTS(v, λ)s. Furthermore, from a type of large set (or over-large set) of MTS(v,λ), we can get the same type of large set (or overlarge set) of DTS(v, λ) and HTS(v, λ).In Chapter 3, the existence problem of the large sets of disjoint pure directed triple systems (LPDTS) is researched. We give some recursiveconstructions and results about the existence. By these results, the existence of LPDTS(v) only depends on the existence of LPDTS(p + 2), where p are odd primes.In Chapter 4, we extend the results of the overlarge sets of three types of oriented triple systems and give some new recursive methods.In Chapter 5, the large sets and the overlarge sets of idempotent quasi-groups are discussed. Especially, we give the existence spectrum of overlarge sets of idempotent symmetric quasigroups.In Chapter 6, we discuss the existence problem of large sets and overlarge sets of three types of mixed triple systems. We give the existence spectrum of LTi(v, λ)(with the unknown order v = 41), OLTi(v}(i = 1,2) and LT3(v, 2λ), OLT3(v,2)(with the unknown order v = 6).更多还原