*m*-Reduction for OOAs

It is shown in [1, Corollary 5.10] that every (linear) ordered orthogonal array OOA(*b*^{m}, *s*, *S*_{b}, *T *, *k*) with *T * > 1 yields a (linear) OOA(*b*^{m−u}, *s*, *S*_{b}, *T *−1, *k*−*u*) for all *u* = 1,…, min{*k*, *s*}. If this propagation rule is applied multiple times, it allows obtaining a (linear) OOA(*b*^{m−u}, *s*, *S*_{b}, *T * – ⌈*u*/*s*⌉, *k*−*u*) for all *u* = 1,…, min{*k*,(*T * – 1)*s*}.

A weaker result (yielding only *T *ʹ = *T *−*u* instead of *T *ʹ = *T * – ⌈*u*/*s*⌉) is given in [2, Theorem 4.4].

This propagation rule cannot be applied to orthogonal arrays because the depth of the resulting OOA is always strictly less than the original depth.

### Construction

The new OOA Aʹ is obtained from A as

*x*

_{1,2},…,

*x*

_{1,T }| ⋯ |

*x*

_{u,2},…,

*x*

_{u,T }|

*x*

_{u+1,1},…,

*x*

_{1,T −1}| ⋯ |

*x*

_{s,1},…,

*x*

_{s,T −1}) :

*∈ A and*

**x***x*

_{1,1}= … =

*x*

_{1,u}= 0},

i.e., by dropping the most significant digits in the first *u* blocks all runs that do not have certain fixed elements for these digits.

### See Also

For

*T*→∞ the corresponding result for nets is obtained.

### References

[1] | Rudolf Schürer.Ordered Orthogonal Arrays and Where to Find Them.PhD thesis, University of Salzburg, Austria, August 2006. |

[2] | William J. Martin and Douglas R. Stinson. A generalized Rao bound for ordered orthogonal arrays and ( t, m, s)-nets.Canadian Mathematical Bulletin, 42(3):359–370, 1999.MR1703696 (2000e:05030) |

### Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.

Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “*m*-Reduction for OOAs.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_OMRed.html