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Nets, (t, s)-Sequences, and Algebraic Geometry

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Random and Quasi-Random Point Sets

Part of the book series: Lecture Notes in Statistics ((LNS,volume 138))

Abstract

The star discrepancy is a classical measure for the irregularity of distribution of finite sets and infinite sequences of points in the s-dimensional unit cube P = [0, 1]8. Point sets and sequences with small star discrepancy in I8 are informally called low-discrepancy point sets, respectively low-discrepancy sequences, in I. It is also customary to speak of sets, respectively sequences, of quasirandom points in 78. Such point sets and sequences play a crucial role in applications of numerical quasi-Monte Carlo methods. In fact, the efficiency of a quasi-Monte Carlo method depends to a significant extent on the quality of the quasirandom points that are employed, i.e., on how small their star discrepancy is. Therefore, it is a matter of considerable interest to devise techniques for the construction of point sets and sequences with as small a star discrepancy as possible. The reader who desires more background on discrepancy theory and quasi-Monte Carlo methods is referred to the books of Hua and Wang [9], Kuipers and Niederreiter [10], and Niederreiter[21], the survey article of Niederreiter [16], and the recent monograph of Drmota and Tichy [4].

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Authors and Affiliations

  1. Institute of Information Processing, Austrian Academy of Sciences, Sonnenfelsgasse 19, Vienna, A-1010, Austria

    Harald Niederreiter & Chaoping Xing

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  1. Harald Niederreiter
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  2. Chaoping Xing
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  1. Institut für Mathematik, Universität Salzburg, Hellbrunner Str. 34, Salzburg, A-5020, Austria

    Peter Hellekalek  & Gerhard Larcher  & 

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Niederreiter, H., Xing, C. (1998). Nets, (t, s)-Sequences, and Algebraic Geometry. In: Hellekalek, P., Larcher, G. (eds) Random and Quasi-Random Point Sets. Lecture Notes in Statistics, vol 138. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1702-2_6

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  • DOI: https://doi.org/10.1007/978-1-4612-1702-2_6

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-98554-1

  • Online ISBN: 978-1-4612-1702-2

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