Constacyclic Codes of Arbitrary Length Over $F_{Q}+Uf_{Q}+Cdots +U^{E1}F_{Q}$

In this article, we shall study the structure of $(a+bu)$constacyclic codes of arbitrary length over the ring$r=f_{q}+uf_{q}+cdots +u^{e1}f_{q}$, where $u^{e}=0$, $q$ is a power of a prime number $p$ and $a,b$ are nonzero elements of $f_{q}$. also we shall find a minimal spanning set for these codes.  %, and completely determine the structure of these codes. for a constacyclic code $c$ we shall determine its minimum hamming distance with some properties of $tor(c)$ as an $a$constacyclic code over $f_{q}$.