Sets of orthogonal hypercubes of class \(r\). (English) Zbl 1232.05034
Summary: A \((d,n,r,t)\)-hypercube is an \(n\times n\times \dots \times n\) (\(d\)-times) array on \(nr\) 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 \(d\geqslant 2r\), 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.
MSC:
| 05B15 | Orthogonal arrays, Latin squares, Room squares |
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