m-Reduction for OOAs

It is shown in [1, Corollary 5.10] that every (linear) ordered orthogonal array OOA(bm, s, Sb, T , k) with T > 1 yields a (linear) OOA(bm−u, s, Sb, 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(bm−u, s, Sb, 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

Aʹ = {(x1,2,…, x1,T  | ⋯ | xu,2,…, xu,T  | xu+1,1,…, x1,T −1 | ⋯ | xs,1,…, xs,T −1)  :  x ∈ A and x1,1 = … = x1,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

References

[1]Rudolf Schürer.
Ordered Orthogonal Arrays and Where to Find Them.
PhD thesis, University of Salzburg, Austria, August 2006. PDF
[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: 2024-09-05. http://mint.sbg.ac.at/desc_OMRed.html

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