环F_(p^k)+uF_(p^k)+u^2F_(p^k)上码是循环码的一个充分必要条件

A Necessary and Sufficient Condition for a Code to be Cyclic Over Ring F_(p^k) + uF_(p^k) + u^2F_(p^k)

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.

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.