Limited Ring Based On The Hamming Weight Of Macwilliams Identity Research

Nowadays, coding theory of finite field has been developed very mature and widely applied in life. With the development of the theory and the advancement of technology, coding theory over finite ring has more important theoretical significant, the most important part is the value of application. Both over finite field or finite ring, different weight enumerators of the linear codes play an important role in studying the coding theory. Weight enumerator is the weight distribution, and it is the useful tool that study weight distribution and structure of codes.In this paper, we studied two different weight enumerators of linear codes and their Mac Williams identities over F2+vF2. First, the complete Lee weight enumerator is defined by Lee weight. Then, we give the mapping τ, and it made finite ring mapping to finite field. And, its C-MacWilliams identity is given. Furthermore, the general MacWilliams identity is given--T-MacWilliams identity of t-Lee weight enumerator. This t-Lee weight enumerator has a parameter t that could change, and the number of parameters of weight enumerator of dual codes are depend on t. We can also get complete Lee weight enumerator when t=1. Second, the concept of the exact weight is introduced and the exact weight enumerator is defined by exact weight. Then we give the mapping δ,ε. And, its E-MacWilliams identity is given. Furthermore, we generalized E-MacWilliams identity to the Multi-dimensions, and get the general MacWilliams identity--M-Mac Williams identity. From the M-Mac William identity of exact weight enumerator of lines codes, we can get its dual codes’ MacWilliams identity of exact weight enumerator, and we can also get dual codes quickly and exactly.