  Working over the field of order 2 we consider those complete caps (maximal
sets of points with no three collinear) which are disjoint from some
codimension 2 subspace of projective space. We derive restrictive conditions
which such a cap must satisfy in order to be complete. Using these conditions
we obtain explicit descriptions of complete caps which do not meet every
hyperplane in at least 5 points. In particular, we determine the set of
cardinalities of all such complete caps in all dimensions.
