We introduce a Gray map from $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2}^{2m}$ and study $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},$ where $u^{2}=0.$ It is proved that the image of a $(1+u)$-constacyclic code length $n$ over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ under the Gray map is a distance-invariant quasi-cyclic code of index $m$ and length $2mn$ over $\mathbb{F}_{2}.$ We also prove that every code of length $2mn$ which is the Gray image of cyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ of length $n$ is permutation equivalent to a binary quasi-cyclic code of index $m.$ Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}.$