## Salzburg Tables

In [1] and [2, Sect. 4.4] a construction for digital (*t*, *m*, *s*)-nets over **F**_{b} based on formal Laurent series over **F**_{b} is proposed. Given polynomials *g*_{1},…, *g*_{s} and *f*, all over **F**_{b}, deg *g*_{i} < *m*, and deg *f* = *m*, we consider the Laurent expansions

*u*

_{w}

^{(i)}

*x*

^{−w}

for *i* = 1,…, *s* and define the elements *c*_{jk}^{(i)} of the generator matrices **C**^{(i)} by *c*_{jr}^{(i)} := *u*_{j+r}^{(i)}. Now **C**^{(1)},…,**C**^{(s)} define a digital (*t*, *m*, *s*)-net with

*t*≤

*m*+ 1 – min(1 + deg

*h*

_{i}),

where the minimum extends over all nonzero *s*-tuples (*h*_{1},…, *h*_{s}) for which *f* divides *g*_{i}*h*_{i}.

Obviously, the quality of the resulting net depends on the polynomials *f*, *g*_{1},…, *g*_{s} used in the construction. Few theoretical results are known about choosing them properly, so one has to resort to a search. A crucial simplification is the restriction to *g*_{i} = *g*^{i−1} mod *f* for some *g* ∈ **F**_{b}[*x*].

A first attempt to find optimal polynomials *f* and *g* was made in [3], however the parameter range was restricted to *b* = 2, *m* ≤ 20 for *s* = 3, 4, and *m* ≤ 10 for larger dimensions.

In [4] the parameter range was extended to *b* = 2, *m* ≤ 25 and *s* ≤ 15. A search was performed over all possible *g*’s with *f* fixed to *f* (*x*) = *x*^{m}, because in this case arithmetic modulo *f* becomes trivial. The tables of *t*-parameters found with this approach became known as “Salzburg Tables”.

In [5] the parameter range was extended to *b* = 2, *m* ≤ 39 and *s* ≤ 50. In addition to that, an irreducible polynomial *f* was used instead of *f* (*x*) = *x*^{m}. This choice complicates determining the *t*-parameter, but leads to lower values.

### References

[1] | Harald Niederreiter. Low-discrepancy point sets obtained by digital constructions over finite fields. Czechoslovak Mathematical Journal, 42:143–166, 1992.MR1152177 (93c:11055) |

[2] | Harald Niederreiter.Random Number Generation and Quasi-Monte Carlo Methods, volume 63 of CBMS-NSF Regional Conference Series in Applied Mathematics.SIAM Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1992. MR1172997 (93h:65008) |

[3] | T. Hansen, Gary L. Mullen, and Harald Niederreiter. Good parameters for a class of node sets in quasi-Monte Carlo integration. Mathematics of Computation, 61:225–234, 1993. |

[4] | Gerhard Larcher, A. Lauß, Harald Niederreiter, and Wolfgang Ch. Schmid. Optimal polynomials for ( t, m, s)-nets and numerical integration of multivariate Walsh series.SIAM Journal on Numerical Analysis, 33(6):2239–2253, 1996.doi:10.1137/S0036142994264705 |

[5] | Wolfgang Ch. Schmid. Improvements and extensions of the “Salzburg Tables” by using irreducible polynomials. In Harald Niederreiter and Jerome Spanier, editors, Monte Carlo and Quasi-Monte Carlo Methods 1998, pages 436–447. Springer-Verlag, 2000. |

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