首先给出了环$\text{bm}R=Z_4+v{Z}_4(v^2=v)$上线性码的Gray映射及其投影映射的性质,得到了环$R$上线性码与通过投影映射得到的线性码的极小{\rm Lee}重量的关系,然后定义了环R上线性码的Gray重量计数器和对称重量计数器,进一步地确定了环$R$上线性码与其对偶码之间关于Gray重量计数器,对称重量计数器和{\rm Lee}重量计数器的MacWilliams恒等式.

In this paper, first we give the propositions of Gray maps and several projections of linear codes over the ring $R=Z_4+v{Z}_4(v^2=v)$ and consider the relationship of minimum {\rm Lee} weight between the linear codes over the ring $R$ and the corresponding codes which are obtained by means of the projections. Next, by defining the Gray weight enumerator and symmetrized weight enumerator, we could establish the MacWilliams identities between linear codes and their dual over the ring $R$ with respect to the Gray weight enumerator, the symmetrized weight enumerator and the {\rm Lee} weight enumerator.