中文摘要：

首先给出了环R=Fp＋vFp＋v2Fp上线性码及其对偶码的结构及其Gray象的性质.定义了环R上线性码的各种重量计数器并讨论了它们之间的关系,特别的,确定了该环上线性码与其对偶码之间关于完全重量计数器的MacWilliams恒等式,利用该恒等式,进一步建立了该环上线性码与其对偶码之间的一种对称形式的MacWilliams恒等式.最后,利用该对称形式的MacWilliams恒等式得到了该环上的Hamming重量计数器和Lee重量计数器的MacWilliams恒等式,利用不同的方法推广了文献[7]中的结果.

英文摘要：

We first give the structures of linear codes and the proposition of their Gray images. A complete weight enumera- tor of linear codes over ring R = Fp ＋ vFp ＋ v2 Fp 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 over R with respect to the complete weight enumerator is given. By using this identity, a syrnmetrized form MacWilliams identitiy between linear codes and their dual over the ring is also established,and MacWilliams identities with respect to Hamming and lee enumerator can be as results of the symmetrized MacWilliams identitiy, we generalize the results in[ 7] using different method.