### Abstract

If the number of runs in a (mixed-level) orthogonal array of strength 2 is specified, what numbers of levels and factors are possible? The collection of possible sets of parameters for orthogonal arrays with N runs has a natural lattice structure, induced by the "expansive replacement" construction method. In particular the dual atoms in this lattice are the most important parameter sets, since any other parameter set for an N-run orthogonal array can be constructed from them. To get a sense for the number of dual atoms, and to begin to understand the lattice as a function of N, we investigate the height and the size of the lattice. It is shown that the height is at most ⌊c(N-1)⌋, where c = 1.4039..., and that there is an infinite sequence of values of N for which this bound is attained. On the other hand, the number of nodes in the lattice is bounded above by a superpolynomial function of N (and superpolynomial growth does occur for certain sequences of values of N). Using a new construction based on "mixed spreads", all parameter sets with 64 runs are determined. Four of these 64-run orthogonal arrays appear to be new.

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

Pages (from-to) | 477-500 |

Number of pages | 24 |

Journal | Journal of Statistical Planning and Inference |

Volume | 102 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2002 |

Externally published | Yes |

### Fingerprint

### Keywords

- Asymmetrical orthogonal array
- Expansive replacement method
- Geometric orthogonal array
- Linear orthogonal array
- Linear programming bound
- Mixed orthogonal array
- Mixed spread
- Tight array

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Applied Mathematics
- Statistics and Probability

### Cite this

*Journal of Statistical Planning and Inference*,

*102*(2), 477-500. https://doi.org/10.1016/S0378-3758(01)00119-7

**The lattice of N-run orthogonal arrays.** / Rains, E. M.; Sloane, N. J A; Stufken, John.

Research output: Contribution to journal › Article

*Journal of Statistical Planning and Inference*, vol. 102, no. 2, pp. 477-500. https://doi.org/10.1016/S0378-3758(01)00119-7

}

TY - JOUR

T1 - The lattice of N-run orthogonal arrays

AU - Rains, E. M.

AU - Sloane, N. J A

AU - Stufken, John

PY - 2002/4/1

Y1 - 2002/4/1

N2 - If the number of runs in a (mixed-level) orthogonal array of strength 2 is specified, what numbers of levels and factors are possible? The collection of possible sets of parameters for orthogonal arrays with N runs has a natural lattice structure, induced by the "expansive replacement" construction method. In particular the dual atoms in this lattice are the most important parameter sets, since any other parameter set for an N-run orthogonal array can be constructed from them. To get a sense for the number of dual atoms, and to begin to understand the lattice as a function of N, we investigate the height and the size of the lattice. It is shown that the height is at most ⌊c(N-1)⌋, where c = 1.4039..., and that there is an infinite sequence of values of N for which this bound is attained. On the other hand, the number of nodes in the lattice is bounded above by a superpolynomial function of N (and superpolynomial growth does occur for certain sequences of values of N). Using a new construction based on "mixed spreads", all parameter sets with 64 runs are determined. Four of these 64-run orthogonal arrays appear to be new.

AB - If the number of runs in a (mixed-level) orthogonal array of strength 2 is specified, what numbers of levels and factors are possible? The collection of possible sets of parameters for orthogonal arrays with N runs has a natural lattice structure, induced by the "expansive replacement" construction method. In particular the dual atoms in this lattice are the most important parameter sets, since any other parameter set for an N-run orthogonal array can be constructed from them. To get a sense for the number of dual atoms, and to begin to understand the lattice as a function of N, we investigate the height and the size of the lattice. It is shown that the height is at most ⌊c(N-1)⌋, where c = 1.4039..., and that there is an infinite sequence of values of N for which this bound is attained. On the other hand, the number of nodes in the lattice is bounded above by a superpolynomial function of N (and superpolynomial growth does occur for certain sequences of values of N). Using a new construction based on "mixed spreads", all parameter sets with 64 runs are determined. Four of these 64-run orthogonal arrays appear to be new.

KW - Asymmetrical orthogonal array

KW - Expansive replacement method

KW - Geometric orthogonal array

KW - Linear orthogonal array

KW - Linear programming bound

KW - Mixed orthogonal array

KW - Mixed spread

KW - Tight array

UR - http://www.scopus.com/inward/record.url?scp=0036531861&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036531861&partnerID=8YFLogxK

U2 - 10.1016/S0378-3758(01)00119-7

DO - 10.1016/S0378-3758(01)00119-7

M3 - Article

VL - 102

SP - 477

EP - 500

JO - Journal of Statistical Planning and Inference

JF - Journal of Statistical Planning and Inference

SN - 0378-3758

IS - 2

ER -