Latin k-hypercubes

We study k dimensional Latin hypercubes of order n. We describe the automorphism groups of the hypercubes and define the parity of a hypercube and relate the parity with the determinant of a permutation hypercube. We determine the parity in the orbits of the automorphism group. Based on this definition of parity we make a conjecture similar to the Alon-Tarsi conjecture. We define an orthogonality relation between hypercubes and we show that a set of mutually orthogonal Latin k dimensional hypercubes correspond to MDS codes and to k+1 dimensional permutation hypercubes satisfying a combinatorial condition.