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Duality theory and propagation rules for higher order nets. (English) Zbl 1229.11103
The higher order nets and sequences are used in quasi-Monte Carlo rules for accurately evaluating high dimensional integrals of smooth functions.
This paper introduces a duality theory for higher order nets whose construction is not necessarily based on linear algebra over finite fields. Furthermore, this paper proves propagation rules which show how to obtain new higher order nets and sequences from existing ones and how the parameters of these point sets propagate under these rules.

MSC:
 11K45 Pseudo-random numbers; Monte Carlo methods 11K36 Well-distributed sequences and other variations
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References:
 [1] Baldeaux, J.; Dick, J.; Pillichshammer, F., A characterisation of higher order nets using Weyl sums and its applications, Unif. distrib. theory, 5, 133-155, (2010) · Zbl 1249.11071 [2] Blackmore, N.; Norton, G.H., Matrix-product codes over $$\mathbb{F}_q$$, Appl. algebra engrg. comm. comput., 12, 477-500, (2001) · Zbl 1004.94034 [3] Dick, J., Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high dimensional periodic functions, SIAM J. numer. anal., 45, 2141-2176, (2007) · Zbl 1158.65007 [4] Dick, J., Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order, SIAM J. numer. anal., 46, 1519-1553, (2008) · Zbl 1189.42012 [5] Dick, J., On quasi-Monte Carlo rules achieving higher order convergence, (), 73-96 · Zbl 1184.65004 [6] Dick, J.; Baldeaux, J., Equidistribution properties of generalized nets and sequences, (), 305-322 · Zbl 1228.65007 [7] Dick, J.; Kritzer, P., Duality theory and propagation rules for generalized digital nets, Math. comp., 79, 993-1017, (2010) · Zbl 1219.11115 [8] Dick, J.; Pillichshammer, F., Digital nets and sequences. discrepancy theory and quasi-Monte Carlo integration, (2010), Cambridge University Press · Zbl 1282.65012 [9] Niederreiter, H., Point sets and sequences with small discrepancy, Monatsh. math., 104, 273-337, (1987) · Zbl 0626.10045 [10] Niederreiter, H., () [11] Niederreiter, H., Construction of $$(t, m, s)$$-nets, (), 70-85 · Zbl 0941.65003 [12] Niederreiter, H.; Pirsic, G., Duality for digital nets and its applications, Acta arith., 97, 173-182, (2001) · Zbl 0972.11066 [13] Niederreiter, H.; Xing, C., Low-discrepancy sequences and global function fields with many rational places, Finite fields appl., 2, 241-273, (1996) · Zbl 0893.11029 [14] G. Pirsic, Embedding theorems and numerical integration of Walsh series over groups, Ph.D. Thesis, University of Salzburg, Austria, 1997. [15] Pirsic, G., Base changes for $$(t, m, s)$$-nets and related sequences, Sitz.ber., oesterr. akad. wiss. math.-nat.wiss. kl. II, 208, 115-122, (1999) · Zbl 1020.11051
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