On Strongly Regular Graphs And Related Linear Codes

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.