## Net from Sequence

Every (*t*, *s*)-sequence in base *b* yields (*t*, *m*, *s* + 1)-nets in base *b* for all *m* ≥ *t*. If the sequence contains points **x**_{0},**x**_{1},… ∈ [0, 1]^{s} with **x**_{i} = (*x*_{i}^{(1)},…, *x*_{i}^{(s)}) and fixed *b*-adic expansion, then the net is given by the point set

*x*

_{i}

^{(1)}⌋

_{m},…,⌊

*x*

_{i}

^{(s)}⌋

_{m},

*i*/

*b*

^{m}) :

*i*= 0,…,

*b*

^{m}– 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 **F**_{b} defined by the *s* ∞×∞ matrices **C**_{1},…,**C**_{s} yields a digital (*t*, *m*, *s* + 1)-net over **F**_{b} for all *m* ≥ *t* defined by the *m*×*m* matrices **C**_{1}ʹ,…,* C*ʹ

_{s},

*ʹ*

**C**_{s+1}, where

*ʹ*

**C**_{i}is the upper left

*m*×

*m*sub-matrix of

**C**_{i}for

*i*= 1,…,

*s*and

*ʹ*

**C**_{s+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