Research On Several Classes Of Codes And Their Applications Over Finite Rings(Fields)

With the rapid development of the algebraic coding and error correcting codes theory,the theoretical basis of the finite ring is more and more important in theory and practical application.Especially,the application of information security is gradually recognized by people.On the basis of theoretical re-search over finite rings,this paper makes a deepgoing research on some linear codes which including the weight enumerators,MacWilliams identities of lin-ear codes,trace codes,constacyclic codes,quadratic residue codes,self-dual quasi-twisted codes,LCD quasi-twisted codes over finite rings and self-dual double(or four)negacyclic codes,self-dual generalized quasi-cyclic codes over finite fields.Details are as follows:1?We study the weight enumerators and MacWilliams identities of linear codes over F2[u,v]/<u2-1,v3-v,uv-uv>.The Gray map,the Lee weight enumerators,the symmetric weight enumerators,Hamming weight enumera-tors and complete weight enumerators over finite rings are defined.We study the relationship between linear codes over rings,the structure of its dual codes and Gray images of linear codes.The MacWilliams identities with respect to the symmetric weight enumerators and complete weight enumerators of linear codes are given.2?The trace codes and its image codes over Fp+uFp+uFp+uvFp are studied.By considering two different defining sets over finite rings,we obtain two families of p-ary two-weight codes and three-weight codes.The weight distribution of these codes are determined by Gaussian sum.Finally,we discuss the application of several families of codes in secret sharing schemes.3?We investigate a special kind of constacyclic codes over rings Z4[u]/<u2-1>and Fp[u,v]/(u2-1,v3-v,uv-vu>.Firstly,we study some very interest-ing properties and generator polynomials of(1 + 2u)-constacyclic codes over Z4[u]/<u2-1>.The Gray map is defined,and we obtain linear codes by using the Magma tools,which are better than existing documents and code table parameters.Secondly,we study the(1-2v2)-constacyclic codes over Fp[u,v]/<u2-1,v3-v,uv-vu)and discuss the structure of the constacyclic codes over the ring by the Chinese Remainder Theorem(CRT).The related properties of the constacyclic codes are studied by the Gray map.4?We study the quadratic residue codes over Fp + uFp + uFp + uvFp +v2Fp+uv2Fp.The quadratic residue codes are defined by using the CRT and idempotent generators.The relationship between the quadratic residue codes and the extended quadratic residue codes is further discussed.5?The self-dual negacirculant codes of index two(or four)are studied.Firstly,we discuss a special kind of factorization of xn+ 1 with n twice an odd prime over Fq.Secondly,we derive the exact enumeration of the self-dual double(or four)-negacirculant codes and discuss the asymptotic performance of the two families of codes.6?We study self-dual and LCD quasi-twisted codes of length 2m over a finite ring Fq[u]/<uk>.Firstly,we derive the exact enumeration formulae of self-dual(or LCD)double ?-circulant codes by considering the special fac-torizations of xm-?.Finally,we get four families of self-dual and LCD quasi-twisted codes for the different decomposition,which can meet modified Gilbert-Varshamov bound on the relative distance of the codes.7?We discuss the existing of the good self-dual generalized quasi-cyclic codes.By using the method of Cubic and Quintic construction,we construct the two classes of binary codes of different length and define two different map-pings to the two classes of linear codes.Finally,the asymptotic performance of long binary self-dual generalized quasi-cyclic codes is studied.