Extracting Embedded Orthogonal Array

Given a (digital) (mk, m, s)-net N in base b with ks, a (linear) orthogonal array A with parameters OA(bm, s, Sb, k) can be constructed. If N is digital over Fb, A is linear and its dual is a linear [s, sm, k + 1]-code over Fb.

For the digital/linear case the result is given explicitly for the first time in [1, Section 1] and [2, Section 3], even though the connection between digital nets and linear codes was already pointed out in [3, Remark 7.13], where the special case for digital (0, m, s)-nets and MDS-codes is considered.

The result for arbitrary nets and OAs is first given in [4] (see also [5, Corollary 9]). The special case with k = 2 can already be found in [2, Theorem 1].

Construction

A is formed based on the leading digit in the b-adic expansion of the coordinates of the points of N. More formally, A is obtained from N as

A = {(ηi−1(⌊bxi⌋))i=1,…, s  :  xN}

with ηi : Sb↔{0,…, b – 1} denoting arbitrary bijections.

If N is digital over Fb, any k of the first row vectors of the s generator matrices of N are linearly independent and form the generator matrix of a linear OA(bm, s,Fb, k).

See Also

References

[1]Harald Niederreiter.
A combinatorial problem for vector spaces over finite fields.
Discrete Mathematics, 96(3):221–228, December 1991.
doi:10.1016/0012-365X(91)90315-S MR1139449 (92j:11150)
[2]Harald Niederreiter.
Orthogonal arrays and other combinatorial aspects in the theory of uniform point distributions in unit cubes.
Discrete Mathematics, 106/107:361–367, September 1992.
doi:10.1016/0012-365X(92)90566-X MR1181933 (94f:11070)
[3]Harald Niederreiter.
Point sets and sequences with small discrepancy.
Monatshefte für Mathematik, 104(4):273–337, December 1987.
doi:10.1007/BF01294651 MR918037 (89c:11120)
[4]Art B. Owen.
Randomly permuted (t, m, s)-nets and (t, s)-sequences.
In Harald Niederreiter and Peter Jau-Shyong Shiue, editors, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, volume 106 of Lecture Notes in Statistics, pages 299–315. Springer-Verlag, 1995.
[5]Gary L. Mullen and Wolfgang Ch. Schmid.
An equivalence between (t, m, s)-nets and strongly orthogonal hypercubes.
Journal of Combinatorial Theory, Series A, 76(1):164–174, October 1996.
doi:10.1006/jcta.1996.0098

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. “Extracting Embedded Orthogonal Array.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_CFromN.html

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