Perfect hypercubes

An (n, d, a, b)-perfect array is a d-dimensional b 1 × b 2 × ⋯ × b d sized n-ary periodic array containing all possible a 1 × a 2 × ⋯ × a d sized n-ary array exactly once as subarray. If a 1 = a 2 = ⋯ = a d 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 = k n with a suitable k ⩾ 1 . Further we illustrate the main theorem constructing 4 and 5-dimensional hypercubes.