Sets of orthogonal hypercubes of class r

  • Authors:

  • John T. Ethier;Gary L. Mullen;Daniel Panario;Brett Stevens;David Thomson

  • Affiliations:

  • Department of Mathematical and Computer Sciences, Metropolitan State College of Denver, Denver, CO 80217, USA;Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA;School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada;School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada;School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2012

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Abstract

A (d,n,r,t)-hypercube is an nxnx...xn (d-times) array on n^r 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=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.