Macwilliams Identities of Linear Codes over the Ring

The Lee weight enumerators and the complete weight enumerators for the linear codes over ring R = F2 + uF2 + vF2 are defined and Gray map ϕ from R n to F 3n 2 is constructed. By proving the fact that the Gray images of the self-dual codes over R are the self-dual codes over F2, and based on the MacWilliams identities for the Hamming weight enumerators of linear codes over F2, the MacWilliams identities for Lee weight enumerators of linear codes over R are given. Further, by introducing a special variable t, the MacWilliams identities for the complete weight enumerators of linear codes over R are obtained. Finally, an example which illustrates the correctness and function of the two MacWilliams identities is provided.