【作者】 何智文；

【导师】 冯涛；

【作者基本信息】 浙江大学 ， 应用数学， 2021， 博士

【摘要】 强正则图和有向强正则图是组合图论中十分重要的两类图,它们与不同领域中的许多有意思的结构有着密切的联系,如有限几何、编码理论以及组合设计理论等等.研究者们通常借助凯莱图对这两类图进行研究和构造,得到了许多不同类型的强正则图和有向强正则图.组合理论中的结合方案和t-设计是近几十年来被大量研究的两种组合结构,它们在图论、编码理论、密码学、通信和统计学领域有着广泛的应用.特别地,强正则图和无定形对称结合方案有着很强的联系,另外t-设计和线性码之间的相互作用的研究也一直是大家感兴趣的课题.本文将通过代数方法构造非交换2-群上的强正则凯莱图和无定形凯莱结合方案,并利用部分和族构造新的有向强正则凯莱图.本文亦致力于研究由一类三元线性码对应的支撑2-设计所生成的线性码.在第一章中,我们简要介绍了本文的研究背景、文献综述以及主要内容.在第二章中,我们首先考虑了 RT2图,Davis-Xiang图和它们对应的无定形交换凯莱结合方案的正则自同构群.接着得到了通过交换正则自同构群构造非交换的正则自同构群的一般性结论,并将它们应用到这两类图以及对应的无定形交换凯莱结合方案中,从而得到了无定形非交换凯莱结合方案.在第三章中,我们利用了局部环对上的部分和族构造了具有新参数的有向强正则图.得到了16个新的有向强正则图无穷类,并给出了一类新的一致部分和族.在第四章中,我们讨论了由Hermitian函数定义的仿射不变三元码.首先计算了这些三元码的最小重量码字所对应的支撑2-设计的关联矩阵.然后证明了由这些关联矩阵的行所生成的线性码是四阶广义Rccd-Mullcr码的子码,并且它们可以得到2-设计.最后,确定了所得的三元线性码的维数,并给出了该码的最小距离的下界.在第五章中,我们总结了本文所涉及的主要工作,并简略叙述了接下来工作的展望.更多还原

【Abstract】 Strongly regular graphs and directed strongly regular graphs are two important kinds of graphs in combinatorial graph theory.They are closely related to many interesting structures in different fields,such as finite geometry,coding theory,combinatorial design theory and so on.By means of Cayley graph,many different types of strongly regular graphs and directed strongly regular graphs have been obtained.The association scheme and the t-design in combinatorial theory are two kinds of combinatorial structures which have been studied extensively in recent decades.They are widely used in graph theory,coding theory,cryptography,communication and statistics.In particular,strong regular graphs have a strong connection with the amorphic Cayley schemes,and the interaction between t-designs and linear codes has been a topic of interest.In this paper,we will construct strongly regular Cayley graph and amorphic Cayley schemes on non-abelian 2-groups by algebraic method,and construct new directed strongly regular Cayley graph by using partial sum families.This paper is also devoted to the study of linear codes generated by supporting 2-designs corresponding to a class of ternary linear codes.In Chapter 1,we will briefly introduce the research background,literature review and the main content of this paper.In Chapter 2,we consider regular automorphism groups of graphs in the RT2 family and the DavisXiang family and amorphic abelian Cayley schemes from these graphs.We derive general results on the existence of non-abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT2 family and Davis-Xiang family and their amorphic abelian Cayley schemes to produce amorphic non-abelian Cayley schemes.In Chapter 3,we construct directed strongly regular graphs with new parameters by using partial sum families with local rings.16 families of new directed strongly regular graphs are obtained and the uniform partial sum families are given.In Chapter 4,we study the affine-invariant ternary codes defined by Hermitian functions.We first compute the incidence matrices of the 2-designs supported by the minimum weight codewords of these ternary codes.Then we show that the linear codes spanned by the rows of these incidence matrices are subcodes of the 4-th order generalized Reed-Muller codes and also hold 2-designs.Finally,we determine the dimension and develop a lower bound on the minimum distance of our ternary linear codes.In chapter 5,we summarize the main work of this paper and briefly describe the future work.更多还原