OOA Folding and Stacking with Additional Row
Let A denote a (linear) ordered orthogonal array OOA(M, s, Sb, T , k) and let u = ⌈⌊k/2⌋/T ⌉ and sʹ = ⌊(s – 1)/u⌋. Then a (linear) OOA Aʹ with parameters OOA(M, sʹ, Sb, T ʹ, k) can be constructed for any T ʹ ≤ (2u + 1)T . If M = bm and T is chosen as T = k (which is always possible, because (2u + 1)T ≥ k), a (digital) (m−k, m, sʹ)-net in base b is obtained.
For T = 1 and A linear the result is due to [1, Theorem 2] and [2, Theorem 1]. For T = 1 and A non-linear it is due to [1, Theorem 6] and [3, Theorem 6.2.1].
Construction
The OOA Aʹ is constructed based on A as follows: Let σ denote a permutation of {1,…, sʹ} without fixed points. Then the ith factor of Aʹ is constructed as
| ( | xi+0sʹ,1,…, xi+0sʹ,T , …, xi+(u−1)sʹ,1,…, xi+(u−1)sʹ,T , | |
| xs,1,…, xs,T , | ||
| xσ(i)+(u−1)sʹ, 1,…, xσ(i)+(u−1)sʹ, T, …, xσ(i)+0sʹ, 1,…, xσ(i)+0sʹ, T | ) |
for i = 1,…, sʹ. Note that the first Tu levels of Aʹ are obtained by u-times folding the first sʹu factors of A. The last factor of A is copied into all factors of Aʹ in levels Tu + 1,…,(u + 1)T . Finally, levels (u + 1)T + 1,…,(2u + 1)T are a factor-wise permuted and level-wise reversed copy of the first Tu levels.
See Also
If u = 1, there is no folding and the propagation rule OOA stacking with additional row is used.
If T = 1 and k is even, no additional row is required and the propagation rule OA folding and stacking is used.
(Part of) construction 17 in [4]
References
| [1] | Wolfgang Ch. Schmid. (t, m, s)-Nets: Digital Construction and Combinatorial Aspects. PhD thesis, University of Salzburg, Austria, 1995. |
| [2] | Kenneth Mark Lawrence, Arijit Mahalanabis, Gary L. Mullen, and Wolfgang Ch. Schmid. Construction of digital (t, m, s)-nets from linear codes. In S. D. Cohen and Harald Niederreiter, editors, Finite Fields and Applications, volume 233 of Lect. Note Series of the London Math. Soc., pages 189–208. Cambridge University Press, 1996. |
| [3] | Kenneth Mark Lawrence. Combinatorial Bounds and Constructions in the Theory of Uniform Point Distributions in Unit Cubes, Connections with Orthogonal Arrays and a Poset Generalization of a Related Problem in Coding Theory. PhD thesis, University of Wisconsin, Madison, 1995. |
| [4] | Andrew T. Clayman, Kenneth Mark Lawrence, Gary L. Mullen, Harald Niederreiter, and Neil J. A. Sloane. Updated tables of parameters of (t, m, s)-nets. Journal of Combinatorial Designs, 7(5):381–393, 1999. doi:10.1002/(SICI)1520-6610(1999)7:5<381::AID-JCD7>3.0.CO;2-S MR1702298 (2000d:05014) |
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. “OOA Folding and Stacking with Additional Row.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_OFoldingStackingWithAdditionalRow.html