D-Dimensional hypercubes and the Euler and MacNeish conjectures

We give a construction for large sets of mutually orthogonal hypercubes of dimensional d given sets of mutually orthogonal latin squares and hypercubes of lower dimension. We also consider d>-2 dimensional versions of the Euler and MacNeish conjectures as well as discussing applications to improved constructions of (t, m, s)-nets, useful in pseudorandom number generation and quasi-Monte-Carlo methods of numerical integration.