Let R be the ring F2m + uF2m , where u2 = 0. We introduce a Gray map from R to F22m and study (1+u)-constacyclic codes over R. It is proved that the image of a (1+u)-constacyclic code length n over R under the Gray map is a distance-invariant binary quasi-cyclic code of index m and length 2mn. We also prove that every code of length 2mn which is the Gray image of cyclic codes over R 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 R.