### Abstract

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

Original language | English (US) |
---|---|

Pages (from-to) | 430-439 |

Number of pages | 10 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 119 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2012 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Journal of Combinatorial Theory. Series A*,

*119*(2), 430-439. https://doi.org/10.1016/j.jcta.2011.10.001

}

*Journal of Combinatorial Theory. Series A*, vol. 119, no. 2, pp. 430-439. https://doi.org/10.1016/j.jcta.2011.10.001

**Sets of orthogonal hypercubes of class r.** / Ethier, John T.; Mullen, Gary Lee; Panario, Daniel; Stevens, Brett; Thomson, David.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Sets of orthogonal hypercubes of class r

AU - Ethier, John T.

AU - Mullen, Gary Lee

AU - Panario, Daniel

AU - Stevens, Brett

AU - Thomson, David

PY - 2012/2/1

Y1 - 2012/2/1

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.

UR - http://www.scopus.com/inward/record.url?scp=80053895196&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80053895196&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2011.10.001

DO - 10.1016/j.jcta.2011.10.001

M3 - Article

VL - 119

SP - 430

EP - 439

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 2

ER -