TY - JOUR

T1 - Sets of orthogonal hypercubes of class r

AU - Ethier, John T.

AU - Mullen, Gary L.

AU - Panario, Daniel

AU - Stevens, Brett

AU - Thomson, David

N1 - Funding Information:
E-mail addresses: jethier@mscd.edu (J.T. Ethier), mullen@math.psu.edu (G.L. Mullen), daniel@math.carleton.ca (D. Panario), brett@math.carleton.ca (B. Stevens), dthomson@math.carleton.ca (D. Thomson). 1 The final three authors are supported in part by NSERC of Canada.

PY - 2012/2

Y1 - 2012/2

N2 - A (d, n, r, t)-hypercube is an n×n×...×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 nd-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.

AB - A (d, n, r, t)-hypercube is an n×n×...×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 nd-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.

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U2 - 10.1016/j.jcta.2011.10.001

DO - 10.1016/j.jcta.2011.10.001

M3 - Article

AN - SCOPUS:80053895196

SN - 0097-3165

VL - 119

SP - 430

EP - 439

JO - Journal of Combinatorial Theory. Series A

JF - Journal of Combinatorial Theory. Series A

IS - 2

ER -