Summary: We study the structure of $(1+u)$-constacyclic codes of an arbitrary length $n$ over the ring $F_{2}+uF_{2}$. We find a set of generators for each $(1+u)$-constacyclic code and its dual. We study the rank of cyclic codes and find their minimal spanning sets. We prove that the Gray image of a $(1+u)$-constacyclic code is a binary cyclic code of length $2n$. We conclude by giving examples of constacyclic codes and their Gray image binary codes. We give a direct construction of a $[12,7,4]$ linear binary cyclic code that match the Hamming distance of the best binary code with length 12 and dimension 7.