m-Reduction from Arbitrarily Large Net

Every (digital) (t, m, s)-net with m > t yields a (digital) (t, m−1, s)-net in the same base.

This propagation rule is due to [1, Lemma 2.8] for general nets and to [2, Lemma 3] for digital nets.

Construction

The new net Nʹ is obtained from N by selecting all points in [0, 1/b)×[0, 1)s−1N and scaling them back to [0, 1]s using an arbitrary affine transformation.

See Also

References

[1]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)
[2]Wolfgang Ch. Schmid and Reinhard Wolf.
Bounds for digital nets and sequences.
Acta Arithmetica, 78(4):377–399, 1997.
[3]Gary L. Mullen, Arijit Mahalanabis, and Harald Niederreiter.
Tables of (t, m, s)-net and (t, s)-sequence parameters.
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 58–86. Springer-Verlag, 1995.
[4]Harald Niederreiter.
Constructions of (t, m, s)-nets and (t, s)-sequences.
Finite Fields and Their Applications, 11(3):578–600, August 2005.
doi:10.1016/j.ffa.2005.01.001 MR2158777 (2006c:11090)

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. “m-Reduction from Arbitrarily Large Net.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_NMRed-inf.html

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