On The Construction Of Linear Codes And Self-dual Codes Over Finite Non-chain Rings

Error-correcting codes are important for the improvement of efficiency and security in information transmission.Constructing codes with good parameters is a fundamental problem in error-correcting codes.With the development of the theory of error-correcting codes over finite non-chain rings,more and more coding scholars begin to focus on the linear codes over the rings.In this paper,we mainly study the structural properties of linear codes over the finite non-chain ring Fq[v]/(vm-v).For example,we study constacyclic codes over the finite non-chain ring Fq[v]/(vm-v),and some self-dual constant cyclic codes are also introduced;We give some lower and upper bounds on the covering radius of linear codes with Chinese Euclidean distance over the finite non-chain ring F2+vF2;We study some results on(α+βv)-constacyclic codes over Fp+vFp,and constructing non-binary quantum codes over finite fields by special constacyclic codes;We consider MacDonald codes over the finite non-chain ring Fp+vFp+v2 Fp and their torsion codes applications in constructing secret sharing schemes and association schemes.The specific contents are as follows:In Chapter 1,we mainly introduce the research background of coding theory,the research significance of the linear codes over finite non-chain fields,the existing research problems at home and abroad.Finally,we summarize the main research results of this paper.In Chapter 2,we mainly study self-dual constacyclic codes over the finite non-chain ring Fq[v]/(vm-v),including Euclidean self-dual constacyclic codes,Hermitian self-dual constacyclic codes and maximal distance separable(MDS)codes of Hermitian self-dual constacyclic codes.We give a necessary condition for constacyclic codes to be Euclidean self-dual and give a necessary and sufficient condition for constacyclic codes to be Hermitian self-dual over the ring Fq[v]/(vm-v).Further,some good self-dual codes are constructed by the Gray map.Especially,a new Hermitian self-dual code overF192 with parameters[16,8,6]is constructed.In Chapter 3,we give some lower and upper bounds on the covering radius of linear codes with Chinese Euclidean distance over the finite non-chain ring F2+vF2,where v2=v.We determine the bounds on covering radius of repetition codes,simplex codes and MacDonald codes over this ring.In Chapter 4,we study some results on constacyclic codes over Fp+vFp,where p is an odd prime and v2=v.Under some special Gray map,the image of the(α+,βv)-constacyclic code over Fp+vFp is a(-α(α+β))-constacyclic code over Fp,where α and β are non-zero elements of Fp satisfying,β=-2α.As an application,some new non-binary quantum codes are obtained.In Chapter 5,we consider MacDonald codes over the finite non-chain ring Fp+vFp+v Fp and the applications of its torsion codes in constructing secret sharing schemes and association schemes,where p is an odd prime andv3=v.In Chapter 6,the main results of the full paper are summarized and we give some interesting questions for the development of the theory of error-correcting codes over finite non-chain rings.