On The Constructions Of Certain Finite Geometric Structures And Codes

Combinatorics is a very attractive subject.Its research objects are discrete.Combinatorial mathematics has many branches,such as graph theory,combinatorial design,finite geometry and so on.It also has a high degree of coincidence with practical branches of mathematics such as coding theory and cryptography.There are many algebraic structures with great properties in combinatorics,which are widely used in information science,communication,storage and so on.Therefore,the construction problems in combinatorics is interesting and meaningful.This paper deals with the constructions of a class of geometric structures in combinatorial mathematics and several classes of useful codes in the field of information.The purpose of this paper is to deal with the construction problems by finite fields,algebraic number theory,groups,characters and other mathematical tools.In Chapter 1,we introduce the research background of the geometric structures and codes we concerned and the main contributions of this paper.In Chapter 2,we determine the equivalence between strong regular graphs and movoids in the symplectic polar space W(2r-1,pe)under certain conditions,and we give the construction of m-ovoids in the symplectic polar space by using the known strong regular graphs.In Chapter 3,we construct LCD codes and linear codes with 1-dimensional hull by using some amorphic association schemes.Furthermore,we find some optimal LCD codes and linear codes with 1-dimensional hull with computer when the parameters are small.In Chapter 4,we construct large cyclic subspace codes from known Sidon spaces and their variants by using the conversion relationship between cyclic subspace codes with certain conditions and Sidon spaces,our construction solves part of the conjuecture of the existence of cyclic subspace codes satisfying certain conditions.Finally,we will introduce our later work about our research.