部分几何差集与部分几何差族的构造

发布时间：2018-03-16 03:13

本文选题：部分几何设计　切入点：部分几何差集　出处：《北京交通大学》2017年硕士论文　论文类型：学位论文

【摘要】：部分几何设计的概念最初是由Bose,Shrikhande和Singhi[7]在1976年提出的.随后,Bose等人[4-6]广泛研究了部分几何设计的代数和组合性质.1980年,在t(1/2)-设计的研究过程中,Neumaier[27]将部分几何设计作为一个子类,称作1(1/2)-设计.2005年,Van Dam和Spence[12]把部分几何设计当作一类关联矩阵具有两个不同奇异值的组合设计进行研究,并给出了一些小参数的部分几何设计.Chai等人[9]利用有限域上的辛几何构造了一些部分几何设计.作为强正则图的推广,Duval[14]在1988年定义了有向强正则图.由于有向强正则图的参数有很多限制,所以有向强正则图是很稀少的.但是构造方法有很多,如区组设计[17],同调代数[22],有限几何[16],正则竞赛图[21],分块矩阵[1]以及凯莱图[15]等.2012年,Brouwer等人[8]指出利用部分几何设计的旗或者反旗做点,可以通过一个给定的部分几何设计构造出两个有向强正则图.从这个角度出发,有向强正则图的构造就转化为部分几何设计的构造.2013年,Olmez[34]引入了部分几何差集的定义,采用的名称是“1(1/2)-差集”,作为部分几何设计的差集版本;并且,他证明了我们可以通过部分几何差集来构造对称的部分几何设计,与通过一般意义下的差集来构造对称2-设计的方式完全相同.进一步地,Nowak[31]等人又提出部分几何差族的概念,这一概念同时对部分几何差集和一般意义下的差族进行了推广;同时,文章指出一个部分几何差族也可以导出一个部分几何设计.最近,Michel[26]构造了几类新的部分几何差集和部分几何差族,大部分构造所用方法与Nowak等人[30,31]类似,其他构造则利用了平面函数.本文主要研究部分几何差集和部分几何差族的一般构造,一共分为四章.第一章,首先介绍了一些与本文有关的基本概念和符号.第二章,我们在直积群上给出了部分几何差集的几个一般构造,通过应用从而得到几类新的部分几何差集;我们的构造统一并推广了Michel[26],Olmez[33]以及Spence[36]的一些已有结果.第三章,我们构造了几类新的部分几何差族.第四章,我们总结了由前两章构造所导出部分几何设计一些新的参数,因此赋予有向强正则图一些新的参数.
[Abstract]:The concept of partial geometric design was first proposed in 1976 by Boseen Shrikhande and Singhi [7]. Subsequently, Bose et al. [4-6] extensively studied the algebraic and combinatorial properties of partial geometric design. In 1980, Neumaier [27] took partial geometric design as a subclass in the course of the study of tl / 2- design. In 2005, Van Dam and Spence [12] studied partial geometric design as a class of combinatorial designs with two different singular values of the correlation matrix. Some partial geometric designs with small parameters. Chai et al. [9] constructed some partial geometric designs by using symplectic geometry over finite fields. As a generalization of strongly regular graphs, Duval [14] defined directed strongly regular graphs in 1988. There are many restrictions on the parameters of regular graphs, So it's rare to have strongly regular graphs, but there are a lot of ways to construct them. For example, block design [17], homology algebra [22], finite geometry [16], regular tournament graph [21], partitioned matrix [1] and Kelet graph [15]. In 2012, Brouwer et al. Two strongly regular graphs can be constructed by a given partial geometric design. From this point of view, the construction of directed strongly regular graphs is transformed into the construction of partial geometric designs. In 2013, Olmez [34] introduced the definition of partial geometric difference sets. The name used is "1 / 1 / 2 / 2 difference set" as a version of partial geometric design, and he proves that we can construct symmetric partial geometric designs from partial geometric difference sets. The concept of partial geometric difference family is put forward by Nowak [31], which extends the concept of partial geometric difference set and difference family in general sense. At the same time, it is pointed out that a partial geometric difference family can also be derived from a partial geometric design. Recently, Michel [26] constructed several new partial geometric difference sets and partial geometric difference families, most of which are similar to those used by Nowak et al. In this paper, we mainly study the general structure of partial geometric difference sets and partial geometric difference families, which are divided into four chapters. In chapter 1, we first introduce some basic concepts and symbols related to this paper. In this paper, we give some general structures of partial geometric difference sets on direct product groups, and obtain some new partial geometric difference sets by application. We unify and generalize some existing results of Michel [26] Olmez [33] and Spence [36]. We construct several new partial geometric difference families. Chapter 4th, we summarize some new parameters of partial geometric design derived from the first two chapters, so we give some new parameters to strongly regular graphs.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O18

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