对环${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}+uv{\mathbb{F}}_{p^m}$上长为$p^e$ 的所有$(\alpha +u\beta )$-循环码进行了分类,其中u^2=0,v^2=0,uv=vu,\alpha,\beta\in\mathbb{F}^\ast_{p^m}$.对于给定的$\alpha$,$\beta$,$e$,我们得到了该环上长为$p^e$的所有$(\alpha+u\beta)$-循环码的计数公式.最后定义了这些常循环码的扭码和剩余码,并利用扭码获得了该环上长为$p^e$的$(\alpha+u\beta)$- 循环码的~Hamming 距离分布.

In this paper, we classify all $(\alpha +\beta u)$-cyclic codes of length $p^e$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}+uv{\mathbb{F}}_{p^m}$, where $u^2=0,v^2=0,uv=vu,\alpha ,\beta \in {\mathbb{F}}_{p^m}^{\ast }$. For any given $\alpha ,\beta ,e$, a formula for the number of these constacyclic codes is obtained. Then, the Torsion codes and Residue codes of those constacyclic codes are also defined. Finally, the Hamming distance  distribution of those constacyclic codes is given with the help of   their Torsion codes.