## 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(*b*^{m}, *s*, *S*_{b}, *k*) can be constructed. If N is digital over **F**_{b}, A is linear and its dual is a linear [*s*, *s*−*m*, *k* + 1]-code over **F**_{b}.

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

*η*

_{i}

^{−1}(⌊

*bx*

_{i}⌋))

_{i=1,…, s}:

*∈ N}*

**x**with *η*_{i} : *S*_{b}↔{0,…, *b* – 1} denoting arbitrary bijections.

If N is digital over **F**_{b}, 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(*b*^{m}, *s*,**F**_{b}, *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: 2015-09-03.
http://mint.sbg.ac.at/desc_CFromN.html