Net from Sequence

Every (t, s)-sequence in base b yields (t, m, s + 1)-nets in base b for all mt. If the sequence contains points x0,x1,… ∈ [0, 1]s with xi = (xi(1),…, xi(s)) and fixed b-adic expansion, then the net is given by the point set

N = {(⌊xi(1)m,…,⌊xi(s)mi/bm)  :  i = 0,…, bm – 1},

where ⌊⋅⌋m denotes the m-digit truncation in base b. This result is due to [1, Lemma 1], based on the result for (t, s)-sequences in the narrow sense in [2, Lemma 5.15]. For base b = 2 see also [3, Theorem 5.2].

Every digital (t, s)-sequence over Fb defined by the s ∞×∞ matrices C1,…,Cs yields a digital (t, m, s + 1)-net over Fb for all mt defined by the m×m matrices C1ʹ,…,Cʹs,Cʹs+1, where Cʹi is the upper left m×m sub-matrix of Ci for i = 1,…, s and Cʹs+1 is the mirrored m×m identity matrix. It is easy to see that the resulting point set is the same as for the method described above.

References

[1]Harald Niederreiter and Chaoping Xing.
Low-discrepancy sequences and global function fields with many rational places.
Finite Fields and Their Applications, 2(3):241–273, July 1996.
doi:10.1006/ffta.1996.0016 MR1398076 (97h:11080)
[2]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)
[3]Ilʹya M. Sobolʹ.
On the distribution of points in a cube and the approximate evaluation of integrals.
U.S.S.R. Computational Mathematics and Mathematical Physics, 7(4):86–112, 1967.

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. “Net from Sequence.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_NFromS.html

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