### Abstract

The theory of (t, m, s)-nets is useful in the study of sets of points in the unit cube with small discrepancy. It is known that the existence of a (0, 2, s)-net in base b is equivalent to the existence of s-2 mutually orthogonal latin squares of order b. In this paper we generalize this equivalence by showing that for t≥0 the existence of a (t, t+2, s)-net in base b is equivalent to the existence of s mutually orthogonal hypercubes of dimension t+2 and order b. Using the theory of hypercubes we obtain upper bounds on s for the existence of such nets. For b a prime power these bounds are best possible. We also state several open problems.

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

Pages (from-to) | 265-273 |

Number of pages | 9 |

Journal | Monatshefte für Mathematik |

Volume | 113 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 1992 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Monatshefte für Mathematik*,

*113*(4), 265-273. https://doi.org/10.1007/BF01301071

}

*Monatshefte für Mathematik*, vol. 113, no. 4, pp. 265-273. https://doi.org/10.1007/BF01301071

**Point sets with uniformity properties and orthogonal hypercubes.** / Mullen, G. L.; Whittle, G.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Point sets with uniformity properties and orthogonal hypercubes

AU - Mullen, G. L.

AU - Whittle, G.

PY - 1992/12/1

Y1 - 1992/12/1

N2 - The theory of (t, m, s)-nets is useful in the study of sets of points in the unit cube with small discrepancy. It is known that the existence of a (0, 2, s)-net in base b is equivalent to the existence of s-2 mutually orthogonal latin squares of order b. In this paper we generalize this equivalence by showing that for t≥0 the existence of a (t, t+2, s)-net in base b is equivalent to the existence of s mutually orthogonal hypercubes of dimension t+2 and order b. Using the theory of hypercubes we obtain upper bounds on s for the existence of such nets. For b a prime power these bounds are best possible. We also state several open problems.

AB - The theory of (t, m, s)-nets is useful in the study of sets of points in the unit cube with small discrepancy. It is known that the existence of a (0, 2, s)-net in base b is equivalent to the existence of s-2 mutually orthogonal latin squares of order b. In this paper we generalize this equivalence by showing that for t≥0 the existence of a (t, t+2, s)-net in base b is equivalent to the existence of s mutually orthogonal hypercubes of dimension t+2 and order b. Using the theory of hypercubes we obtain upper bounds on s for the existence of such nets. For b a prime power these bounds are best possible. We also state several open problems.

UR - http://www.scopus.com/inward/record.url?scp=0040453061&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040453061&partnerID=8YFLogxK

U2 - 10.1007/BF01301071

DO - 10.1007/BF01301071

M3 - Article

AN - SCOPUS:0040453061

VL - 113

SP - 265

EP - 273

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

SN - 0026-9255

IS - 4

ER -