Cube designs

A cube design of order v, or a CUBE(v), is a decomposition of all cyclicly oriented quadruples of a v-set into oriented cubes. A CUBE(v) design is unoriented if its cubes can be paired so that the cubes in each pair are related by reflection through the center. A cube design is degenerate if it has repeated points on one of its cubes, otherwise it is nondegenerate. We show that a nondegenerate CUBE(v) design exists for all integers v ≥ 8, and that an unoriented nondegenerate CUBE(v) design exists if and only if v ≥ 8 and v ≡ 0, 1, 2 or 3 (mod 8).