Research On The Chinese Product Of Cyclic Codes And The MacWilliams Identities For Linear Codes Over Finite Rings

A great deal of attention has been paid to codes over finite rings from the 1990s, because scholars have found that certain nonlinear codes (such as Kerdock codes, Preparata codes, Goethal codes) over finite fields can be constructed from linear codes over finite rings via Gray maps. Since then, studying coding theory over finite rings is a hot topic. In this thiese, we study the Chinese product of cyclic codes over Z k, ( k = ( i∏=s 1pi)m, with the condition that the code length n can not be divided by pi ). We also study the MacWilliams identities for Lee and Euclidean weights of linear codes over Galois rings; and we study the MacWilliams identities with respect to any partition and support weight for linear codes over Z_p~2( p is a prime). The details are given as follows:1. We describe the Chinese Remainder Theorem for studying cyclic and dual cyclic codes over Z k, (k = ( i∏=s 1pi)m, with the condition that the code length n can not be divided by pi ). A necessary and suffcient condition of the existence of nontrivial self-dual cyclic codes is given. The upper bound of minimum distance of cyclic codes is also obtained. Furthermore, we determine the minimum generators of such cyclic codes.2. The m ? ply Lee and Euclidean weight enumerators of linear codes over Z k are defned, and then we obtain a new kind of MacWilliams identities for the Lee and Euclidean weights of such codes. Furthermore, we regard Galois ring GR ( p e, m ) as a free Z pe module of rank m , then we discuss the relationships between the Lee and Euclidean weight enumerators of linear codes over GR ( p e, m ), which have generator matrices over Z peand the m ? ply Lee and Euclidean weight enumerators of linear codes over Z pe. At last, we get the Lee and Euclidean weights MacWilliams identities of linear codes over GR ( p e, m ).3. The generalized Lee weight enumerators for linear codes of length n over Z 4are defined. It is given that the generalized Lee weight MacWilliams identities for linear codes of length n over Z 4. The equivalent form of generalized Lee weight enumerators for linear codes of length n over Z 4are defined. Correspondingly, the equivalent form of generalized Lee weight MacWilliams identities for linear codes of length n over Z 4are obtained.4. When we interpret CU as restriction codes of C , we give an equation which connects the weight distribution of a code with those of its restrictions to sets of a given size; when we interpret C U as subcodes of C , we get a set of MacWilliams identities between any partition of the coordinate set of a Z_p~2-linear code with type ( p 2)k and that of its dual are established.5. We also study support weight and the generalized Plotkin bound of linear codes over Z_p~2, and give the generalized Plotkin bound for homogenous weight of linear codes over Z_p~2.