Research On Linear Codes Over Several Finite Rings And Their Application

In this paper, we study the MacWilliams identities and quadratic residue codes and N-weight codes over some finite rings. The details are given as follows:1、We investigate the MacWilliams identities for linear codes over Fp+ vFp+v2Fp. We first give the structures of linear codes and the proposition of their Gray images. A complete weight enumerator of linear codes over ring Fp+vF2+v2Fp is defined; we define some weight enumerators of linear codes and their dual codes, and then discuss the relations between them. A MacWilliams identity between linear codes and their dual codes with respect to the complete weight enumerator is given. By using this identity, a symmetrized MacWilliams identity between linear codes and their dual over the ring is also established, and MacWilliams identities with respect to Hamming and Lee weight enumerators can be as results of the symmetrized MacWilliams identity.2、The MacWilliams identities with respect to RT metric for linear codes over Mn×s(Fp+vFp+v2Fp) and Mn×s(Fp+uFp+vFp+uvFp) are studied, respectively. The Lee complete p weight enumerator and the exact complete p weight enumerator over the ring are defined, and the MacWilliams identities with respect to RT metric for these two weight enumerators of linear codes over the two rings are obtained, respectively.3、We study quadratic residue codes over Fp+vFp+v2Fp. By using Chinese Remainder Theorem and generating idempotents of quadratic residue codes over Fp, we define quadratic residue codes over Fp+vFp+v2Fp. Next, we discuss the properties of these codes and their extended codes. Finally, we search for some good codes from quadratic residue codes over Fp+vFp+v2Fp.4、We study simplex codes of type a over Z4+uZ4, and define Gray weight and Gray map over the non-chain ring. Then we define type of codes over the ring in terms of the relation between Gray weight and Hamming weight. We also consider its weight distributions. By the generator matrix of simplex codes, we give the definition of MacDonald codes, and discuss the weight distributions about Torsion codes of MacDonald codes. Finally, we obtain a result, i.e. Torsion codes of MacDonald codes is linear codes of type α over Z4. What is more, it is a two-Lee weight code.5、We consider the Mac Williams identities about Gray weight enumerator over the ring S[v]/<v2+v>, where S=F2+uF2 (u2= 0). Then we present a necessary condition of a linear code to be a N-Gray codes over the non-chain ring, where N=1 or 2. Moreover we discuss the issue how to construct N-Gray codes.