中文摘要：

近年来，剩余类环Zm上的编码理论有很大的发展，人们通过Gray映射建立了比较系统的Zm环上的编码理论。文章定义了对于n=n1p^δ-，，环Zp^k＋1^1^n到环Zp^1^pn^k的Gray映射，给出了该映射的几个性质，并由此得出了Zp^k＋1环上的指数为p^δt长为n-p^δn1的quasi—cyclic码与Zp环上的quasi—cyclic码的一一对应关系，这里的t｜n1，（n1，p）：1，从而环Zp^k＋1上的quasi-cyclic码可以看作是环Zp上的quasi-cyclic码，也即看作是有限素域上的quasi-cyclic码。

英文摘要：

Codes over the finite field has been widely applied in communications. In recent years, codes over residue rings have improved, many systemic results over the residue ring Zm were given via the Gray map. In this paper ,the Gray map from Zp^k＋1^1^n toZp^1^pn^k, is defined, and one of the propositions of this map is given . We show that a quasi-cyclic code of length n = n1p^δ with index p^δt over Zp^k＋1 is uniquely equivalent to a quasi-cyclic code of length p^kn with index p^k＋δ-1 t over Zp, where t ｜ n1, （ n1, p ） = 1. Then the quasi-cyclic code over the ring Zp^k＋1 can be taken as the quasi-cyclic code over the ring Zp which is also a finite prime field.