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A set of \(n+1\) orthogonal squares of order \(n\) is known to be equivalent to a complete set of \(n-1\) mutually orthogonal Latin squares of order \(n\) together with canonical row and column squares. In this note, we show that this equivalence does not extend to orthogonal hypercubes of dimensions \(d > 2\) by providing examples of affine designs that can be represented by complete sets of type \(0\) orthogonal hypercubes but not by complete sets of orthogonal Latin hypercubes together with canonical hypercubes that generalize the row and column squares in the case where \(d = 2\). These examples also clarify the relationship between affine designs and orthogonal hypercubes that generalize the classical equivalence between affine planes and complete sets of MOLS.
We conclude with the statement of a number of conjectures regarding some open questions.