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Working over the field of order 2 we consider those complete caps 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.

Complete Caps Projective Space Which are Disjoint from a Subspace of Codimension Two