Extracting Embedded Orthogonal Array
Given a (digital) (m−k, m, s)-net N in base b with k ≤ s, 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, s−m, 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
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
Generalization for arbitrary OOAs
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: 2024-09-05.
http://mint.sbg.ac.at/desc_CFromN.html