{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T03:07:28Z","timestamp":1775099248053,"version":"3.50.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2014,9,30]],"date-time":"2014-09-30T00:00:00Z","timestamp":1412035200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2015,10]]},"DOI":"10.1007\/s10208-014-9226-8","type":"journal-article","created":{"date-parts":[[2014,9,29]],"date-time":"2014-09-29T18:55:27Z","timestamp":1412016927000},"page":"1245-1278","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":["Construction of Interlaced Scrambled Polynomial Lattice Rules of Arbitrary High Order"],"prefix":"10.1007","volume":"15","author":[{"given":"Takashi","family":"Goda","sequence":"first","affiliation":[]},{"given":"Josef","family":"Dick","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2014,9,30]]},"reference":[{"key":"9226_CR1","unstructured":"J. Baldeaux, Higher order nets and sequences, Ph.D. thesis, The University of New South Wales, 2010."},{"key":"9226_CR2","doi-asserted-by":"crossref","unstructured":"J. Baldeaux and J. Dick, A construction of polynomial lattice rules with small gain coefficients. Numer. Math. 119 (2011), 271\u2013297.","DOI":"10.1007\/s00211-011-0385-0"},{"key":"9226_CR3","doi-asserted-by":"crossref","unstructured":"J. Baldeaux, J. Dick, J. Greslehner and F. Pillichshammer, Construction algorithms for higher order polynomial lattice rules. J. Complexity 27 (2011), 281\u2013299.","DOI":"10.1016\/j.jco.2010.06.002"},{"key":"9226_CR4","doi-asserted-by":"crossref","unstructured":"J. Baldeaux, J. Dick, G. Leobacher, D. Nuyens and F. Pillichshammer, Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules. Numer. Algorithms 59 (2012) 403\u2013431.","DOI":"10.1007\/s11075-011-9497-y"},{"key":"9226_CR5","doi-asserted-by":"crossref","unstructured":"H. E. Chrestenson, A class of generalized Walsh functions. Pacific J. Math. 5 (1955) 17\u201331.","DOI":"10.2140\/pjm.1955.5.17"},{"key":"9226_CR6","doi-asserted-by":"crossref","unstructured":"J. Dick, Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions. SIAM J. Numer. Anal. 45 (2007) 2141\u20132176.","DOI":"10.1137\/060658916"},{"key":"9226_CR7","doi-asserted-by":"crossref","unstructured":"J. Dick, Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46 (2008) 1519\u20131553.","DOI":"10.1137\/060666639"},{"key":"9226_CR8","doi-asserted-by":"crossref","unstructured":"J. Dick, On quasi-Monte Carlo rules achieving higher order convergence, Springer, Berlin, 2009, pp. 73\u201396.","DOI":"10.1007\/978-3-642-04107-5_5"},{"key":"9226_CR9","doi-asserted-by":"crossref","unstructured":"J. Dick, Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands. Ann. Statist. 39 (2011) 1372\u20131398.","DOI":"10.1214\/11-AOS880"},{"key":"9226_CR10","doi-asserted-by":"crossref","unstructured":"J. Dick and M. Gnewuch, Optimal randomized changing dimension algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition. J. Approx. Theory 184 (2014) 111\u2013145.","DOI":"10.1016\/j.jat.2014.04.014"},{"key":"9226_CR11","doi-asserted-by":"crossref","unstructured":"J. Dick, F.Y. Kuo, F. Pillichshammer and I.H. Sloan, Construction algorithms for polynomial lattice rules for multivariate integration. Math. Comp. 74 (2005) 1895\u20131921.","DOI":"10.1090\/S0025-5718-05-01742-4"},{"key":"9226_CR12","doi-asserted-by":"crossref","unstructured":"J. Dick, G. Leobacher, and F. Pillichshammer, Construction algorithms for digital nets with low weighted star discrepancy. SIAM. J. Numer. Anal. 43 (2005) 76\u201395.","DOI":"10.1137\/040604662"},{"key":"9226_CR13","doi-asserted-by":"crossref","unstructured":"J. Dick and F. Pillichshammer, Strong tractability of multivariate integration of arbitrary high order using digitally shifted polynomial lattice rules. J. Complexity 23 (2007) 436\u2013453.","DOI":"10.1016\/j.jco.2007.02.001"},{"key":"9226_CR14","doi-asserted-by":"crossref","unstructured":"J. Dick and F. Pillichshammer, Digital nets and sequences: discrepancy theory and quasi-Monte Carlo integration. Cambridge University Press, Cambridge, 2010.","DOI":"10.1017\/CBO9780511761188"},{"key":"9226_CR15","doi-asserted-by":"crossref","unstructured":"J. Dick, I. H. Sloan, X. Wang and H. Wo\u017aniakowski, Good lattice rules in weighted Korobov spaces with general weights. Numer. Math. 103 (2006) 63\u201397.","DOI":"10.1007\/s00211-005-0674-6"},{"key":"9226_CR16","unstructured":"H. Faure, Discr\u00e9pances de suites associ\u00e9es \u00e0 un syst\u00e8me de num\u00e9ration (en dimension s). Acta Arith. 41 (1982) 337\u2013351."},{"key":"9226_CR17","doi-asserted-by":"crossref","unstructured":"F. J. Hickernell, The mean square discrepancy of randomized nets. ACM Trans. Modeling Comput. Simul. 6 (1996) 274\u2013296.","DOI":"10.1145\/240896.240909"},{"key":"9226_CR18","unstructured":"N. M. Korobov, The approximate computation of multiple integrals\/approximate evaluation of repeated integrals. Dokl. Akad. Nauk SSSR 124 (1959) 1207\u20131210."},{"key":"9226_CR19","doi-asserted-by":"crossref","unstructured":"P. Kritzer and F. Pillichshammer, Constructions of general polynomial lattices for multivariate integration. Bull. Austral. Math. Soc. 76 (2007) 93\u2013110.","DOI":"10.1017\/S0004972700039496"},{"key":"9226_CR20","unstructured":"L. Kuipers and H. Niederreiter, Uniform distribution of sequences. Pure and Applied Mathematics. Wiley-Interscience, New York-London-Sydney, 1974."},{"key":"9226_CR21","doi-asserted-by":"crossref","unstructured":"G. Larcher, A. Lauss, H. Niederreiter and W. Ch. Schmid, Optimal polynomials for $$(t, m, s)$$ ( t , m , s ) -nets and numerical integration of multivariate Walsh series. SIAM. J. Numer. Anal. 33 (1996) 2239\u20132253.","DOI":"10.1137\/S0036142994264705"},{"key":"9226_CR22","doi-asserted-by":"crossref","unstructured":"C. Lemieux and P. L\u2019Ecuyer, Randomized polynomial lattice rules for multivariate integration and simulation. SIAM. J. Sci. Comput. 24 (2003) 1768\u20131789.","DOI":"10.1137\/S1064827501393782"},{"key":"9226_CR23","doi-asserted-by":"crossref","unstructured":"J. Matou\u0161ek, On the $$L_2$$ L 2 discrepancy for anchored boxes. J. Complexity 14 (1998) 527\u2013556.","DOI":"10.1006\/jcom.1998.0489"},{"key":"9226_CR24","doi-asserted-by":"crossref","unstructured":"T. M\u00fcller-Gronbach, E. Novak and K. Ritter, Monte Carlo-Algorithmen. (German) Springer-Lehrbuch. Springer, Heidelberg, 2012.","DOI":"10.1007\/978-3-540-89141-3"},{"key":"9226_CR25","doi-asserted-by":"crossref","unstructured":"H. Niederreiter, Low-discrepancy and low-dispersion sequences. J. Number Theory 30 (1988) 51\u201370.","DOI":"10.1016\/0022-314X(88)90025-X"},{"key":"9226_CR26","doi-asserted-by":"crossref","unstructured":"H. Niederreiter, Random number generation and quasi-Monte Carlo methods. in: CBMS-NSF Series in Applied Mathematics, vol. 63, SIAM, Philadelphia, 1992.","DOI":"10.1137\/1.9781611970081"},{"key":"9226_CR27","unstructured":"H. Niederreiter, Low-discrepancy point sets obtained by digital constructions over finite fields. Czechoslovak Math. J. 42 (1992) 143\u2013166."},{"key":"9226_CR28","doi-asserted-by":"crossref","unstructured":"H. Niederreiter and C. P. Xing, Rational points on curves over finite fields: theory and applications. London Mathematical Society Lecture Note Series, 285. Cambridge University Press, Cambridge, 2001.","DOI":"10.1017\/CBO9781107325951"},{"key":"9226_CR29","doi-asserted-by":"crossref","unstructured":"E. Novak, Deterministic and stochastic error bounds in numerical analysis. Lecture Notes in Mathematics, 1349. Springer-Verlag, Berlin, 1988.","DOI":"10.1007\/BFb0079792"},{"key":"9226_CR30","doi-asserted-by":"crossref","unstructured":"E. Novak and H. Wo\u017aniakowski, Tractability of multivariate problems. Vol. 1: Linear information. EMS Tracts in Mathematics, 6. European Mathematical Society (EMS), Z\u00fcrich, 2008.","DOI":"10.4171\/026"},{"key":"9226_CR31","doi-asserted-by":"crossref","unstructured":"E. Novak and H. Wo\u017aniakowski, Tractability of multivariate problems. Volume II: Standard information for functionals. EMS Tracts in Mathematics, 12. European Mathematical Society (EMS), Z\u00fcrich, 2010.","DOI":"10.4171\/084"},{"key":"9226_CR32","doi-asserted-by":"crossref","unstructured":"D. Nuyens and R. Cools, Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comp. 75 (2006) 903\u2013920.","DOI":"10.1090\/S0025-5718-06-01785-6"},{"key":"9226_CR33","doi-asserted-by":"crossref","unstructured":"D. Nuyens and R. Cools, Fast component-by-component construction, a reprise for different kernels. Springer, Berlin, 2006, pp. 373\u2013387.","DOI":"10.1007\/3-540-31186-6_22"},{"key":"9226_CR34","doi-asserted-by":"crossref","unstructured":"A. B. Owen, Randomly permuted $$(t, m, s)$$ ( t , m , s ) -nets and $$(t, s)$$ ( t , s ) -sequences. in Monte Carlo and quasi-Monte Carlo Methods in Scientific Computing, Springer, New York, 1995, pp. 299\u2013317.","DOI":"10.1007\/978-1-4612-2552-2_19"},{"key":"9226_CR35","doi-asserted-by":"crossref","unstructured":"A. B. Owen, Monte Carlo variance of scrambled net quadrature. SIAM. J. Numer. Anal. 34 (1997) 1884\u20131910.","DOI":"10.1137\/S0036142994277468"},{"key":"9226_CR36","doi-asserted-by":"crossref","unstructured":"A. B. Owen, Scrambled net variance for integrals of smooth functions. Ann. Statist. 25 (1997) 1541\u20131562.","DOI":"10.1214\/aos\/1031594731"},{"key":"9226_CR37","doi-asserted-by":"crossref","unstructured":"A. B. Owen, Variance with alternative scramblings of digital nets. ACM Trans. Model. Comp. Simul. 13 (2003) 363\u2013378.","DOI":"10.1145\/945511.945518"},{"key":"9226_CR38","doi-asserted-by":"crossref","unstructured":"I. H. Sloan and A. V. Reztsov, Component-by-component construction of good lattice rules. Math. Comp. 71 (2002) 263\u2013273.","DOI":"10.1090\/S0025-5718-01-01342-4"},{"key":"9226_CR39","doi-asserted-by":"crossref","unstructured":"I. H. Sloan and H. Wo\u017aniakowski, When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals? J. Complexity 14 (1998) 1\u201333.","DOI":"10.1006\/jcom.1997.0463"},{"key":"9226_CR40","unstructured":"I. M. Sobol\u2019, The distribution of points in a cube and approximate evaluation of integrals. Zh. Vycisl. Mat. i Mat. Fiz. 7 (1967) 784\u2013802."},{"key":"9226_CR41","doi-asserted-by":"crossref","unstructured":"J. L. Walsh, A closed set of normal orthogonal functions. Amer. J. Math. 45 (1923) 5\u201324.","DOI":"10.2307\/2387224"}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-014-9226-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10208-014-9226-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-014-9226-8","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T11:05:14Z","timestamp":1559127914000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10208-014-9226-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,9,30]]},"references-count":41,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2015,10]]}},"alternative-id":["9226"],"URL":"https:\/\/doi.org\/10.1007\/s10208-014-9226-8","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,9,30]]}}}