中文摘要：

记环R=F_p~k＋uF_p~k＋u~2F_p~k,定义了一个从R~n到F_p~k~（2np~k）的Gray映射.利用Gray映射的性质,研究了环R上任意长循环码.证明了环R上任意长码是循环码当且仅当它的Gray象是F_p~k上的准循环码.特别的,环R上的线性循环码的Gray象是F_p~k上的线性准循环码.

英文摘要：

Let R = F_p~k ＋ uF_p~k ＋ u~2F_p~k,a Gray map from R~n to F_（p~k）~（2np~k） is defined.Base on the property of Gray map,cyclic codes of arbitrary length over R are studied.It is proved that a code of arbitrary length over R is a cyclic code if and only if its Gray image is a quasi-cyclic code over F_p~k.In particular,the Gray image of a linear cyclic code over R is a linear quasicyclic code over F_p~k.