Duality theory and propagation rules for generalized digital nets

Dick, Josef; Kritzer, Peter

doi:10.1090/S0025-5718-09-02315-1

Duality theory and propagation rules for generalized digital nets
HTML articles powered by AMS MathViewer

by Josef Dick and Peter Kritzer PDF

Math. Comp. 79 (2010), 993-1017 Request permission

Abstract:

Digital nets are used in quasi-Monte Carlo algorithms for approximating high dimensional integrals over the unit cube. Hence one wants to have explicit constructions of digital nets of high quality. In this paper we consider the so-called propagation rules for digital nets, which state how one can obtain a new digital net of different size from existing digital nets. This way one often can generate digital nets of higher quality than were previously known. Here we generalize existing propagation rules for classical digital nets to generalized digital nets as introduced by Dick.

References

Dick, J., Baldeaux, J.: Equidistribution properties of generalized nets and sequences. To apear in: L’Ecuyer, P. and Owen, A. (eds.): Monte Carlo and Quasi-Monte Carlo Methods 2008, 2010.
Jürgen Bierbrauer, Yves Edel, and Wolfgang Ch. Schmid, Coding-theoretic constructions for $(t,m,s)$-nets and ordered orthogonal arrays, J. Combin. Des. 10 (2002), no. 6, 403–418. MR 1932120, DOI 10.1002/jcd.10015
Tim Blackmore and Graham H. Norton, Matrix-product codes over $\Bbb F_q$, Appl. Algebra Engrg. Comm. Comput. 12 (2001), no. 6, 477–500. MR 1873271, DOI 10.1007/PL00004226
Josef Dick, Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions, SIAM J. Numer. Anal. 45 (2007), no. 5, 2141–2176. MR 2346374, DOI 10.1137/060658916
Josef Dick, Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order, SIAM J. Numer. Anal. 46 (2008), no. 3, 1519–1553. MR 2391005, DOI 10.1137/060666639
Josef Dick, Peter Kritzer, Friedrich Pillichshammer, and Wolfgang Ch. Schmid, On the existence of higher order polynomial lattices based on a generalized figure of merit, J. Complexity 23 (2007), no. 4-6, 581–593. MR 2372015, DOI 10.1016/j.jco.2006.12.003
Harald Niederreiter, Point sets and sequences with small discrepancy, Monatsh. Math. 104 (1987), no. 4, 273–337. MR 918037, DOI 10.1007/BF01294651
Harald Niederreiter, Random number generation and quasi-Monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1172997, DOI 10.1137/1.9781611970081
Harald Niederreiter, Constructions of $(t,m,s)$-nets, Monte Carlo and quasi-Monte Carlo methods 1998 (Claremont, CA), Springer, Berlin, 2000, pp. 70–85. MR 1849843
Harald Niederreiter, Constructions of $(t,m,s)$-nets and $(t,s)$-sequences, Finite Fields Appl. 11 (2005), no. 3, 578–600. MR 2158777, DOI 10.1016/j.ffa.2005.01.001
Harald Niederreiter and Ferruh Özbudak, Matrix-product constructions of digital nets, Finite Fields Appl. 10 (2004), no. 3, 464–479. MR 2067609, DOI 10.1016/j.ffa.2003.11.004
Harald Niederreiter and Gottlieb Pirsic, Duality for digital nets and its applications, Acta Arith. 97 (2001), no. 2, 173–182. MR 1824983, DOI 10.4064/aa97-2-5
Harald Niederreiter and Chaoping Xing, Nets, $(t,s)$-sequences, and algebraic geometry, Random and quasi-random point sets, Lect. Notes Stat., vol. 138, Springer, New York, 1998, pp. 267–302. MR 1662844, DOI 10.1007/978-1-4612-1702-2_{6}
Harald Niederreiter and Chaoping Xing, Constructions of digital nets, Acta Arith. 102 (2002), no. 2, 189–197. MR 1889629, DOI 10.4064/aa102-2-7
R. Schürer, W.Ch. Schmid: MinT—the database of optimal net, code, OA, and OOA parameters. Available at: http://mint.sbg.ac.at (\today).