Abstract
The most powerful current methods of constructing low-discrepancy point sets for quasi-Monte Carlo applications employ the theory of (t, m, s)-nets. This paper gives a survey of this theory and of the construction methods that are based on it. Some new results are also included.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adams, M.J., Shader, B.L.: A construction for (t,m,s)-nets in base q. SIAM J. Discrete Math. 10 (1997) 460 – 468
Auer, R.: Ray class fields of global function fields with many rational places. Preprint, 1998
Clayman, A.T., Lawrence, K.M., Mullen, G.L., Niederreiter, H., Sloane, N.J.A.: Updated tables of parameters of (t,m,s)-nets. J. Combinatorial Designs, to appear
Clayman, A.T., Mullen, G.L.: Improved (t,m,s)-net parameters from the Gilbert-Varshamov bound. Applicable Algebra Engrg. Comm. Comp. 8(1997) 491–496
Edel, Y., Bierbrauer, J.: Construction of digital nets from BCH-codes. Monte Carlo and Quasi-Monte Carlo Methods 1996 (H. Niederreiter et al., eds.), Lecture Notes in Statistics, Vol. 127, pp. 221–231, Springer, New York, 1998
Faure, H.: Discrepance de suites associées à un système de numération (en dimension s). Acta Arith. 41 (1982) 337 - 351
Hansen, T., Mullen, G.L., Niederreiter, H.: Good parameters for a class of node sets in quasi-Monte Carlo integration. Math. Comp. 61(1993) 225–234
Larcher, G., Lauß, A., Niederreiter, H., Schmid, W.Ch.: Optimal polyno-mials for (t,m,s)-nets and numerical integration of multivariate Walsh series. SIAM J. Numer. Analysis 33(1996) 2239–2253
Larcher, G., Niederreiter, H., Schmid, W.Ch.: Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration. Monatsh. Math. 121(1996) 231–253
Lawrence, K.M.: A combinatorial characterization of (t,m,s)-nets in base b. J. Combinatorial Designs 4(1996) 275–293
Lawrence, K.M.: Construction of (t,m,s)-nets and orthogonal arrays from binary codes. Finite Fields Appl., to appear
Lawrence, K.M., Mahalanabis, A., Mullen, G.L., Schmid, W.Ch.: Construction of digital (t,m,s)-nets from linear codes. Finite Fields and Applications (S. Cohen and H. Niederreiter, eds.), London Math. Soc. Lecture Note Series, Vol. 233, pp. 189–208, Cambridge Univ. Press, Cambridge, 1996
Laywine, C.F., Mullen, G.L.: Discrete Mathematics Using Latin Squares. Wiley, New York, 1998
Laywine, C.F., Mullen, G.L., Whittle, G.: D-dimensional hypercubes and the Euler and MacNeish conjectures. Monatsh. Math. 119(1995) 223–238
Mullen, G.L., Mahalanabis, A., Niederreiter, H.: Tables of (t,m,s)-net and (t,s)-sequence parameters. Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (H. Niederreiter and P.J.-S. Shiue, eds.), Lecture Notes in Statistics, Vol. 106, pp. 58–86, Springer, New York, 1995
Mullen, G.L., Schmid, W.Ch.: An equivalence between (t,m,s)-nets and strongly orthogonal hypercubes. J. Combinatorial Theory Ser. A 76(1996) 164–174
Mullen, G.L., Whittle, G.: Point sets with uniformity properties and orthogonal hypercubes. Monatsh. Math. 113(1992) 265–273
Niederreiter, H.: Point sets and sequences with small discrepancy. Monatsh. Math. 104(1987) 273–337
Niederreiter, H.: Low-discrepancy and low-dispersion sequences. J. Number Theory 30(1988) 51–70
Niederreiter, H.: Low-discrepancy point sets obtained by digital constructions over finite fields. Czechoslovak Math. J. 42(1992) 143–166
Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia, 1992
Niederreiter, H.: Orthogonal arrays and other combinatorial aspects in the theory of uniform point distributions in unit cubes. Discrete Math. 106/107(1992) 361–367
Niederreiter, H.: Nets, (t,s)-sequences, and algebraic curves over finite fields with many rational points. Proc. International Congress of Mathematicians (Berlin, 1998), Documenta Math. Extra Volume ICM III (1998) 377–386
Niederreiter, H., Xing, C.P.: Low-discrepancy sequences obtained from algebraic function fields over finite fields. Acta Arith. 72(1995) 281–298
Niederreiter, H., Xing, C.P.: Low-discrepancy sequences and global function fields with many rational places. Finite Fields Appl. 2(1996) 241–273
Niederreiter, H., Xing, C.P.: Quasirandom points and global function fields. Finite Fields and Applications (S. Cohen and H. Niederreiter, eds.), London Math. Soc. Lecture Note Series, Vol. 233, pp. 269–296, Cambridge Univ. Press, Cambridge, 1996
Niederreiter, H., Xing, C.P.: The algebraic-geometry approach to low-discrepancy sequences. Monte Carlo and Quasi-Monte Carlo Methods 1996 (H. Niederreiter et al., eds.), Lecture Notes in Statistics, Vol. 127, pp. 139–160, Springer, New York, 1998
Niederreiter, H., Xing, C.P.: Nets, (t,s)-sequences, and algebraic geometry. Random and Quasi-Random Point Sets (P. Hellekalek and G. Larcher, eds.), Lecture Notes in Statistics, Vol. 138, pp. 267–302, Springer, New York, 1998
Schmid, W.Ch.: (t,m,s)-nets: digital construction and combinatorial aspects. Dissertation, University of Salzburg, 1995
Schmid, W.Ch.: Shift-nets: a new class of binary digital (t,m,s)-nets. Monte Carlo and Quasi-Monte Carlo Methods 1996 (H. Niederreiter et al., eds.), Lecture Notes in Statistics, Vol. 127, pp. 369–381, Springer, New York, 1998
Schmid, W.Ch.: Improvements and extensions of the “Salzburg Tables” by using irreducible polynomials. This volume
Schmid, W.Ch., Wolf, R.: Bounds for digital nets and sequences. Acta Arith. 78(1997) 377–399
Soboi’, I.M.: The distribution of points in a cube and the approximate evaluation of integrals (Russian). Zh. Vychisl. Mat. i Mat. Fiz. 7(1967) 784–802
Tezuka, S.: Polynomial arithmetic analogue of Halton sequences. ACM Trans. Modeling and Computer Simulation 3(1993) 99–107
Tezuka, S.: Uniform Random Numbers: Theory and Practice. Kluwer Academic Pubi., Boston, 1995
Tezuka, S., Tokuyama, T.: A note on polynomial arithmetic analogue of Halton sequences. ACM Trans. Modeling and Computer Simulation 4(1994) 279–284
Xing, C.P., Niederreiter, H.: A construction of low-discrepancy sequences using global function fields. Acta Arith. 73(1995) 87–102
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Niederreiter, H. (2000). Constructions of (t, m, s)-Nets. In: Niederreiter, H., Spanier, J. (eds) Monte-Carlo and Quasi-Monte Carlo Methods 1998. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59657-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-59657-5_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66176-4
Online ISBN: 978-3-642-59657-5
eBook Packages: Springer Book Archive