On Double Cyclic Codes Over Galois Rings

With the rapid development of modern information and communication technology,information security technology is playing an increasingly important role in society.Improving the efficiency and reliability of information transmission is a development goal in the field of information security.Error-correcting coding theory is an important way to improve the efficiency and reliability of information transmission.In recent years,with the development of coding theory over finite fields,coding scholars begin to pay attention to coding theory over finite rings and its applications.Double cyclic codes are an important class of linear codes,which not only have a good algebraic structure,but also have an important application in engineering.In this thesis,the algebraic structure and enumeration of double cyclic codes over Galois rings are studied,and their applications in constructing quantum error-correcting codes and Hermitian self-dual codes are discussed.The specific contents are as follows:In Chapter 1,we introduce the research background and significance,the overseas and domestic research status of double cyclic codes over Galois rings.Moreover,the main research contents of this thesis are introduced.In Chapter 2,we give some basic results of algebra over Galois rings and cyclic codes over Galois rings which lay the foundation for the subsequent writing of this thesis.In Chapter 3,we discuss the algebraic structures of double cyclic codes over Galois rings,including the generator polynomials,the minimum generating sets,and the Hermitian dual codes.In Chapter 4,based on the structures of double cyclic codes over Galois rings,we determine the enumeration formulas of double cyclic codes over Galois rings.In chapter 5,based on the algebraic structure of double cyclic codes over Galois rings,some new quantum error-correcting codes with new parameters and Hermitian self-dual double cyclic codes with good performance are constructed by using Hermitian selforthogonal double cyclic codes and Hermitian self-dual double cyclic codes over Galois rings.In chapter 6,we summary the main research results of this thesis,and puts forward the problems that need further research.