Sets of orthogonal hypercubes of class r

A ( d , n , r , t ) -hypercube is an n × n × ⋯ × n (d-times) array on n r symbols such that when fixing t coordinates of the hypercube (and running across the remaining d − t coordinates) each symbol is repeated n d − 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 ⩾ 2 r , 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.