Cube Designs. Issue 5 (9th February 2015) - Linek, Václav; Soicher, Leonard H.; Stevens, Brett
Abstract: 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 ) . A degenerate example of a CUBE( v ) design is also given for each integer v ≥ 4 .