## OA Folding and Stacking

Let A denote a (linear) orthogonal array OA(*M*, *s*, *S*_{b}, *k*) with even strength *k* and let *s*ʹ = ⌊2*s*/*k*⌋. Then a (linear) ordered orthogonal array OOA(*M*, *s*ʹ, *S*_{b}, *T *, *k*) can be constructed for any *T * ≤ *k*. If *M* = *b*^{m} and *T * is chosen as *T * = *k*, a (digital) (*m*−*k*, *m*, *s*ʹ)-net in base *b* is obtained.

The result in the linear case is due to [1, Theorem 2] and [2, Theorem 1], the non-linear case to [1, Theorem 6] and [3, Theorem 6.2.1].

### Construction

The OOA Aʹ is constructed based on A as follows: Let *u* = *k*/2 and let *σ* denote a permutation of {0,…, *s*ʹ – 1} without fixed points. Then

*(*

**y***) :*

**x***∈ A}*

**x**with

*y*

_{i,l}(

*x*

_{1},…,

*x*

_{s}) =

*x*

_{(i−1)u+l}for

*l*≤

*u*

and

*y*

_{i,l}(

*x*

_{1},…,

*x*

_{s}) =

*x*

_{σ(i−1)u+2u+1-l}for

*l*>

*u*

and *i* = 1,…, *s*ʹ.

### See Also

If

*k*is odd or if A is an OOA with depth*T*> 1, OOA folding and stacking with additional row has to be used.(Part of) Construction 17 in [4]

### References

[1] | Wolfgang Ch. Schmid.(.t, m, s)-Nets: Digital Construction and Combinatorial AspectsPhD 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. “OA Folding and Stacking.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_OFoldingStacking.html