Constacyclic Codes Over Finite Non-Chain Rings And Their Applications

With the advent of the information age,quantum computation and quantum communication have become a hot research topic.Like the classical theory of errorcorrecting codes,error correction is one of the necessary guarantees for the realization of quantum computation and quantum communication.As an important means to improve the efficiency of quantum information transmission,quantum coding theory has developed rapidly.In recent decades,based on the coding theory over finite fields,coders have constructed a large number of quantum error-correcting codes with good error correction performance using classical codes over finite fields.In the past decade,with the development of coding theory over finite rings,the construction of new quantum error-correcting codes based on linear codes over finite rings has attracted widespread attention of coders.As an important class of linear codes,constacyclic codes are an important source of constructing quantum error-correcting codes with superior performance.Constacyclic codes over finite rings have more complex algebraic structures than those over finite fields,which provides more flexibility for constructing new quantum error-correcting codes.In this paper,we mainly study the algebraic structure of several classes constacyclic codes over finite non-chain rings and their applications in the construction of quantum error-correcting codes.The thesis is organized as follows.In Chapter 1,we introduce the research background and significance,the overseas and domestic research status of constructing quantum error-correcting codes based on constacyclic codes over finite non-chain rings.Moreover,the summary of main research results of this thesis is introduced.In Chapter 2,we introduce the basic knowledge required in this paper,including the basic theory of algebra,the theory of error-correcting codes over finite fields,the basic concept and construction theory of quantum error-correcting codes.In Chapter 3,we study the algebraic structure and properties of constacyclic codes over the finite non-chain ring R=Fq+uFq+vFq+uvFq,propose a new method to construct quantum synchronizable codes based on constacyclic codes over R,and obtain the conditions under which the constructed quantum synchronizable codes have the ability to optimally correct synchronization errors.In Chapter 4,we study the algebraic structure and properties of the constacyclic codes over the finite non-chain ring Rs-1=Fq+v1Fq+…+vs-1Fq,give the important properties of the generator matrix of the Gray image of the constacyclic codes over the ring Rs-1,and use cyclic codes and constacyclic codes over Rs-1 to construct maximal entanglement entanglement-assisted quantum error-correcting codes,and maximal entanglement and maximum distance separable entanglement-assisted quantum error-correcting codes.In Chapter 5,based on the Hermitian dual-containing constacyclic codes over finite non-chain R1=Fp2+v1Fp2,we use the defining set of constacyclic codes and the cyclotomic coset theory to construct maximum distance separable quantum codes and subsystem codes.In Chapter 6,we summary the main results of this thesis and list several open problems of quantum error-correcting codes.