Sets of orthogonal hypercubes of class r

A (d,n,r,t)-hypercube is an n×n×⋯×n (d-times) array on nr symbols such that when fixing t coordinates of the hypercube (and running across the remaining d−t coordinates) each symbol is repeated nd−r−t times. We introduce a new parameter, r, representing the class of the hypercube. When r=1, this provides the usual definition of a hypercube and when d=2 and r=t=1 these hypercubes are Latin squares. If d⩾2r, then the notion of orthogonality is also inherited from the usual definition of hypercubes. This work deals with constructions of class r hypercubes and presents bounds on the number of mutually orthogonal class r hypercubes. We also give constructions of sets of mutually orthogonal hypercubes when n is a prime power.