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        {\bf Wolfgang Ch. Schmid} \footnotemark[1] \newline
        \footnotesize
        Department of Mathematics \newline
        University of Salzburg \newline
        Hellbrunnerstr. 34 \newline
        A-5020 Salzburg, Austria \newline
        e-mail: {\tt Wolfgang.Schmid@sbg.ac.at} \newline
        {\tt http://www.mat.sbg.ac.at/people/schmidw.html}

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\ \par

\footnotetext[1]{
      Research supported by the Austrian Science Foundation (FWF),
      project P12441 MAT
                }


\vspace*{0.2cm}
{\sc Title}

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        { \bf On the Exact Determination of the\\
         Quality Parameter of Digital Sequences and\\
         Several Applications}\\
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{\sc Abstract} \ \par

The quality of quasi-Monte Carlo methods mainly depends on the
distribution properties of the underlying (deterministic) point set.
The theory of digital nets and sequences provides a method for the
construction of extremely well distributed point sets in the $s$-dimensional
unit cube.

One of the most common methods for the construction of a digital net
(finite point set) is the use of sequences: every digital
$(t_s ,s)$-sequence provides a digital $(t,m,s+1)$-net for all integers
$m\geq t$ with $t=t_s $.
The value $t$ indicates the quality of the point set and should be as
small as possible.
For the optimal parameter $t$ of the net we clearly have $t \leq t_s $.

We will report on an efficient algorithm for the exact determination
of the optimal quality parameter of digital nets,
and we will show results for the nets derived from
Sobol' and Niederreiter sequences
which are widely used in many  applications of
quasi-Monte Carlo methods, such as physics, technical sciences,
financial mathematics, and many more.

Additionally this algorithm is the basis for several other net
constructions of best quality.
We have to mention the ``Shift--Nets'', or
the improvements and extensions of the ``Salzburg Tables''
which were carried out quite recently.

In another recent project we have defined a new, more subtle, quality
parameter for digital nets and digital sequences. We will give
a short survey of this ``investigation of a new quality parameter''.
A refined version of the above mentioned algorithm again can be used
for concrete computer calculations.

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