| Finite geometry is an important branch in combinatorics mathematics. It provides plentiful sources for graph theory, combinatorics design, coding theory, etc.. The research of finite geometries is an important problem with great theoretic significance and strong application background. The theories and methods of finite geometries play an important role in mathematical statistics, operations research, information theory and computer science. (α,β)-geometries are incidence structures satisfying special codi-tions. The research of (α,β)-geometries can be traced back to the year 1963. In 1963, Bose introduced the concept of partial geometries (partial geometries are a class of (α,β)-geometries) when he studied the relationship between strongly regular graphs and PBD-designs. Currently, theories of partial geometries are abundant, while that of general (α,β)-geometries are much fewer. The research of another special case of (α,β)-geometries -semipartial geometries is very active in recent 10 years. During the research of (α,β)-geometries, mathematicians do not only find some incidence structures with good properties and reveal the essences of this class of incidence structures, but also found and introduce many new theories and methods, while these theories and methods have brought activities into other subjects. By finite geometries is a useful method when one studys linear codes. When one studies the minimum leghth bounds of linear codes, finite geometry is a powerful tool. People also tried to construct good codes from geometry structures. In recent years, mathematicians and computer experts have constructed a group of LDPC codes from geometry structures. Experiments show that the LDPC codes constructed from partial geometries behave well in minimum distance, girth, bit-error rate, etc.. We expect to know about the properties of more general (α,β)-geometries, and study its application in information science. This thesis is devoted to the embeddings, constructions and applications of (α,β)-geometries. This thesis is divided into six chapters.The first chapter is devoted to the summarization of the dissertation. We disscuss the development of the subject, including the history and current situations of it, methods used and difficulties facing us. We also give the main results of this thesis.In the second chapter, we talk about the fully embeddings of (1,β)- geometries in AG(3,q). We prove that when q > 2, a (1, q)—geometry can be fully embedded in AG(3, q) if and only if it is a linear representation. In further, we talk about the case of strongly regular (1,β)—geometries. We also give the necessary and sufficient conditions for a family of (1,β)—geometries fully embedded in AG(3,q) being linear representations when 2 <β< q.In the third chapter, we introuduce the concept of net-inducible strongly regular (1,β)-geometries (β> 2). We describle the properties of a class of minimal net-inducible (1,β) -geometries.In the fourth chapter, we construct a class of (α,β)—geometries using a method from group theory. We show that the theory of these strongly regular (α,β)-geometries is almost equivalent to the study of strongly regular (α,β)-reguli if G is abelian.In the fifth chapter, we construct a group of LDPC codes from semipartial geometries. Experiments show that these LDPC codes perform well.In the sixth chapter, we describe the minimum length bounds of a class of linear codes by methods of finite geometries.
|
PDF Full Text Request