Parallelism Of Lines In Finite Projective Space

With the rapid development of science and technology,network coding has become an emerging technology in recent years.Since the discovery of the widespread application of subspace codes in network coding in 2008,many scholars have focused on them,and subspace codes and subspace design have also developed rapidly.Constant-dimensional codes are a special kind of subspace codes.It is a code composed of subspaces with equal dimensions as codewords.It is closely related to the problem we are considering.This paper considers lines of parallelism in finite projective geometry,that is,dividing lines into disjoint spreads,so that each line appears exactly once in each spread.This paper considers the direct construction method of lines of parallelism,for which we mainly use two mappings,cyclic shift mapping and Frobenius mapping.The two mappings are used to divide the lines in projective space.The automorphism group formed by them provides great convenience in the process of finding solutions.Use the construction of spread and parallelism,we found some specific parameters(n,p)corresponding to the lines of parallelism through computer search.The specific content is divided as follows:In chapter 1,we introduce the background knowledge of network coding,the important applications of subspace coding and constant-dimensional codes,and then we introduce some basic definitions,newest results of the maximum number of disjoint spreads in a finite projective space and some known parallelism.In chapter 2,we introduce the basic knowledge of finite fields and the construction methods,which provide a theoretical basis for subsequent computer searches.Then we present some concepts,basic properties and counting theorems of projective spaces on finite fields,as well as the relationship between vector space and projective space.In chapter 3,we study the direct construction method of lines of parallelism in finite projective space.Firstly,the cyclic shift map,the Frobenius map and the cyclotomic coset are defined,and the characteristics of the two mappings applying on subspaces are introduced.Secondly,the necessary conditions of the direct construction of disjoint spread are presented,the properties of the second type of base line and the third type of generating line,which are closely related to it,are studied.In chapter 4,based on the construction method introduced in the previous article,we list the solutions of lines of parallelism in PG(n,p)for specific parameters through computer search.The main parameters are the parameters(p,n)? {(2,5),(2,7),(3,5),(5,3)}.Then we present the construction method from disjoint spread to disjoint Steiner system and the results of a series of disjoint Steiner systems.In chapter 5,we summarize the main content of this article and discusses what can be further studied and improved.