On the classification of nonsingular 2x2x2x2 hypercubes

As a first step in the classification of nonsingular 2x2x2x2 hypercubes up to equivalence, we resolve the case where the base field is finite and the hypercubes can be written as a product of two 2x2x2 hypercubes. (Nonsingular hypercubes were introduced by D. Knuth in the context of semifields. Where semifields are related to hypercubes of dimension 3, this paper considers the next case, i.e., hypercubes of dimension 4.) We define the notion of ij-rank (with 1 a parts per thousand currency sign i < j a parts per thousand currency sign 4) and prove that a hypercube is the product of two 2x2x2 hypercubes if and only if its 12-rank is at most 2. We derive a 'standard form' for nonsingular 2x2x2x2 hypercubes of 12-rank less than 4 as a first step in the classification of such hypercubes up to equivalence. Our main result states that the equivalence class of a nonsingular 2x2x2x2 hypercube M of 12-rank 2 depends only on the value of an invariant delta (0)(M) which derives in a natural way from the Cayley hyperdeterminant det(0) M and another polynomial invariant det M of degree 4. As a corollary we prove that the number of equivalence classes is (q + 1)/2 or q/2 depending on whether the order q of the field is odd or even.