Perfect hypercubes

An (n, d, a, b)-perfect array is a d-dimensional b1×b2×⋯×bd sized n-ary periodic array containing all possible a1×a2×⋯×ad sized n-ary array exactly once as subarray. If a1=a2=⋯=ad and term double cube are used. If d⩾4, then the double cube is called double hypercube. We prove the existence of (N, d, a, b)-perfect double cubes for arbitrary d⩾1,a⩾2 and n⩾2, where N=kn with a suitable k⩾1. Further we illustrate the main theorem constructing 4 and 5-dimensional hypercubes.