Research On Linear Codes Over Several Finite Communicative Rings

Based on the theory of finite rings,this dissertation is mainly research on MacWilliams identities of linear codes with respect to RT metric;constructing linear codes with few weights over finite rings,and the asymptotic performance for additive constacyclic codes over finite fields.Specific contents are given as follows:1?We study the MacWilliams identities with respect to the RT metric over the finite non-chain ring R1 = F1[v]/(vk-v).First of all,we define the Gray map for the ring R1,and give a partition.Then use the Hadamaxd trans-form,we get the MacWilliams identities with respect to the RT metric for the Lee complete ? weight enumerator and the exact complete ? weight enumera-tor over Mn×s(R1).In addition,some examples are presented to illustrate the obtained results.2?We study the trace codes over finite chain ring Fp +uFp.In the case of choosing defining set L,their Lee weight distribution is computed by using Gauss sums.Then use the distance-preserving Gray map,we get two classes of few weights codes,i.e.,· if m is even,the image code ?(C(m,p))is a three-weight code over Fp;if m is odd and p? 3(mod 4),the image code ?(C(m,p))is a two-weight code over Fp.In particular,we have proved that the obtained two-weight codes meet the Griesmer bound with equality.At the end of this chapter,we give the dual distance of the trace codes,and discuss the minimal property of the codewords in the image codes.3?On the basis of the previous chapter,by choosing three different defin-ing sets,several classes of three-weight codes and two-weight codes for the homogeneous metric over the chain ring R3=F3[u]/(um),are constructed.These codes are defined as trace codes.Their homogeneous weight distribu-tions are computed by using exponential sums.It is noteworthy that when k=2,the codes obtained in Chapter 4 is not a special case of this chapter.In particular,in the two-weight case,we give some conditions for the optimality of their Gray images by using the Griesmer bound.Their dual homogeneous distance is also given.The codewords of these codes are shown to be minimal for inclusion of supports,a fact favorable to an application to secret sharing schemes.4?Long quasi-twisted codes of fixed index>1 have been shown recent-ly to be asymptotically good(A.Alahmadi,C.Giineri,H.Shoaib,P.Sole,2017).In this chapter,we build on a natural map between quasi-twisted codes of index l over Fq and additive constacyclic codes over Fql.Then we can show that additive constacyclic codes over alphabets of non-square order are asymptotically good.