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Point sets and sequences with small discrepancy

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Abstract

A systematic theory of a class of point sets called (t, m, s)-nets and of a class of sequences called (t, s)-sequences is developed. On the basis of this theory, point sets and sequences in thes-dimensional unit cube with the smallest discrepancy that is currently known are constructed. Various connections with other areas arise in this work, e.g. with orthogonal latin squares, finite projective planes, finite fields, and algebraic coding theory. Applications of the theory of (t, m, s)-nets to two methods for pseudorandom number generation, namely the digital multistep method and the GFSR method, are presented. Several open problems, mostly of a combinatorial nature, are stated.

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References

  1. Chen, W. W. L.: On irregularities of distribution II. Quart. J. Math. (2)34, 257–279 (1983).

    Article  MATH  Google Scholar 

  2. Dénes, J., Keedwell, A. D.: Latin Squares and Their Applications. New York: Academic Press. 1974.

    MATH  Google Scholar 

  3. Faure, H.: Discrépances de suites associées à un système de numération (en dimension un). Bull. Soc. Math. France109, 143–182 (1981).

    MATH  MathSciNet  Google Scholar 

  4. Faure, H.: Discrépance de suites associées à un système de numération (en dimensions). Acta Arith.41, 337–351 (1982).

    MATH  MathSciNet  Google Scholar 

  5. Hall, M.: Combinatorial Theory. Waltham: Blaisdell. 1967.

    MATH  Google Scholar 

  6. Halton, J. H.: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math.2, 84–90 (1960); Berichtigung, ibid., 196.

    Article  MATH  MathSciNet  Google Scholar 

  7. Hammersley, J. M.: Monte Carlo methods for solving multivariable problems. Ann. New York Acad. Sci.86, 844–874 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  8. Hlawka, E.: Zur angenäherten Berechnung mehrfacher Integrale. Mh. Math.66, 140–151 (1962).

    Article  MATH  MathSciNet  Google Scholar 

  9. Hlawka, E.: Uniform distribution modulo 1 and numerical analysis. Compositio Math.16, 92–105 (1964).

    MATH  MathSciNet  Google Scholar 

  10. Hua, L. K., Wang, Y.: Applications of Number Theory to Numerical Analysis. Berlin-Heidelberg-New York: Springer. 1981.

    MATH  Google Scholar 

  11. Knuth, D. E.: The Art of Computer Programming, Vol. 2: Seminumerical Algorithms. 2nd ed. Reading: Addison-Wesley. 1981.

    MATH  Google Scholar 

  12. Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. New York: Wiley-Interscience. 1974.

    MATH  Google Scholar 

  13. Lidl, R., Niederreiter, H.: Finite Fields. Reading: Addison-Wesley. 1983.

    MATH  Google Scholar 

  14. Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge: Cambridge Univ. Press. 1986.

    MATH  Google Scholar 

  15. Macwilliams, F. J., Sloane, N. J. A.: The Theory of Error-Correcting Codes. Amsterdam: North-Holland. 1977.

    MATH  Google Scholar 

  16. Mcdonald, B. R.: Finite Rings with Identity. New York: Dekker. 1974.

    MATH  Google Scholar 

  17. Muir, T.: History of Determinants, Vol. IV. London: Macmillan. 1923.

    Google Scholar 

  18. Niederreiter, H.: Pseudo-random numbers and optimal coefficients. Adv. in Math.26, 99–181 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  19. Niederreiter, H.: Quasi-Monte Carlo methods and pseudo-random numbers. Bull. Amer. Math. Soc.84, 957–1041 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  20. Niederreiter, H.: Existence of good lattice points in the sense of Hlawka. Mh. Math.86, 203–219 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  21. Niederreiter, H.: The serial test for pseudo-random numbers generated by the linear congruential method. Numer. Math.46, 51–68 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  22. Niederreiter, H.: Multidimensional numerical integration using pseudorandom numbers. Math. Programming Study27, 17–38 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  23. Niederreiter, H.: Low-discrepancy point sets. Mh. Math.102, 155–167 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  24. Niederreiter, H.: Pseudozufallszahlen und die Theorie der Gleichverteilung. Sitzungsber. Österr. Akad. Wiss. Math.-Naturwiss. Kl.195, 109–138 (1986).

    MATH  MathSciNet  Google Scholar 

  25. Niederreiter, H.: A statistical analysis of generalized feedback shift register pseudorandom number generators. SIAM J. Sci. Statist. Comp. (To appear.)

  26. Niederreiter, H.: The serial test for digitalk-step pseudorandom numbers. Math. J. Okayama Univ. (To appear.)

  27. Roth, K. F.: On irregularities of distribution. Mathematika1, 73–79 (1954).

    Article  MATH  MathSciNet  Google Scholar 

  28. Schmidt, W. M.: Irregularities of distribution. VII. Acta Arith.21, 45–50 (1972).

    MATH  MathSciNet  Google Scholar 

  29. Sobol', I. M.: The distribution of points in a cube and the approximate evaluation of integrals. Zh. Vychisl. Mat. i Mat. Fiz.7, 784–802 (1967). (Russian.)

    MathSciNet  Google Scholar 

  30. Sobol', I. M.: Multidimensional Quadrature Formulas and Haar Functions. Moscow: Nauka. 1969. (Russian.)

  31. Tezuka, S.: On optimal GFSR pseudorandom number generators. Submitted to Math. Comp.

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Authors and Affiliations

  1. Kommission für Mathematik, Österreichische Akademie der Wissenschaften, Dr. Ignaz-Seipel-Platz 2, A-1010, Wien, Austria

    Harald Niederreiter

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  1. Harald Niederreiter
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The author wants to thank the Universität Hannover (West Germany) for its hospitality during the period when most of this research was carried out. The results of this paper were presented at the Colloquium on Irregularities of Partitions at Fertöd (Hungary), July 7–11, 1986.

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Niederreiter, H. Point sets and sequences with small discrepancy. Monatshefte für Mathematik 104, 273–337 (1987). https://doi.org/10.1007/BF01294651

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  • DOI: https://doi.org/10.1007/BF01294651

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