## Shift-Nets

In [1] Schmid defines a special class of digital (*t*, *m*, *m*)-nets, the so-called shift nets. These nets are completely defined by their first coordinate, therefore the number of possible generator matrices is reduced significantly. Few theoretical results are known about the existence of shift nets, but using computer search it turns out that shift nets of high or even optimal quality exist for many sizes *m*. Furthermore, the special structure of shift nets simplifies the test for an assumed strength *k*. These properties allow a randomized and even a full search for shift nets in a parameter range that is completely out of reach for arbitrary digital nets. For small bases the obtained parameters cannot be reached by any other known method.

For *b* = 2 and 1 ≤ *m* ≤ 22, *t*-values and generator matrices are given in the original article [1]. In [2] Schmid and Schürer generalize shift nets to arbitrary prime power bases and refine the search algorithm compared with the original approach.

In [3, Chapter 2] the search algorithm is again improved significantly, allowing to obtain good shift nets for *m* ≤ 51 for *b* = 2, *m* ≤ 38 for *b* = 3, *m* ≤ 33 for *b* = 4, *m* ≤ 25 for *b* = 5, *m* ≤ 20 for *b* = 7, *m* ≤ 17 for *b* = 8, and *m* ≤ 15 for *b* = 9. Generator matrices for *b* = 2 and *b* = 3 can also be found in [3, Appendix B].

### See Also

Construction 21 in [4].

### References

[1] | Wolfgang Ch. Schmid. Shift-nets: a new class of binary digital ( t, m, s)-nets.In Harald Niederreiter, Peter Hellekalek, Gerhard Larcher, and Peter Zinterhof, editors, Monte Carlo and Quasi-Monte Carlo Methods 1996, volume 127 of Lecture Notes in Statistics, pages 369–381. Springer-Verlag, 1998. |

[2] | Wolfgang Ch. Schmid and Rudolf Schürer. Shift-nets and Salzburg tables: Power computing in number-theoretical numerics. In Helmut Efinger and Andreas Uhl, editors, Scientific Computing in Salzburg--Festschrift on the Occasion of Peter Zinterhofs 60th Birthday, volume 189 of books@ocg.at, pages 175–184. Oesterreichische Computer Gesellschaft, 2005. |

[3] | Rudolf Schürer.Ordered Orthogonal Arrays and Where to Find Them.PhD thesis, University of Salzburg, Austria, August 2006. |

[4] | Andrew T. Clayman, Kenneth Mark Lawrence, Gary L. Mullen, Harald Niederreiter, and Neil J. A. Sloane. Updated tables of parameters of ( t, m, s)-nets.Journal of Combinatorial Designs, 7(5):381–393, 1999.doi:10.1002/(SICI)1520-6610(1999)7:5<381::AID-JCD7>3.0.CO;2-S MR1702298 (2000d:05014) |

### 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. “Shift-Nets.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_NShift.html