A Complete Classification of Digital (0,m, 3)-Nets and Digital (0, 2)-Sequences in Base 2

Hofer, Roswitha; Suzuki, Kosuke

doi:10.2478/udt-2019-0004

Abstract

We give a complete classification of all matrices C₁,C₂,C₃ ∈ $C_{3} \in 𝔽_{2}^{m \times m}$ {C_3} \in \mathbb{F}_2^{m \times m} which generate a digital (0,m, 3)-net in base 2 and a complete classification of all matrices C₁,C₂ ∈ $C_{2} \in 𝔽_{2}^{ℕ \times ℕ}$ {C_2} \in \mathbb{F}_2^{\mathbb{N} \times \mathbb{N}} which generate a digital (0, 2)-sequence in base 2.

References

[1] DICK, J.—PILLICHSHAMMER, F.: Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge, 2010.10.1017/CBO9780511761188
Search in Google Scholar Back to article
[2] FAURE, H.: Discrépance de suites associées à un système de numération (en dimension s),Acta Arith. 41 (1982), 337–351.10.4064/aa-41-4-337-351
Search in Google Scholar Back to article
[3] FAURE, H.—TEZUKA, S.: Another random scrambling of digital (t, s)-sequences. In: Monte Carlo and Quasi-Monte Carlo Methods 2000, (K. T. Fang et. al, eds.), Springer-Verlag, Berlin, 2002, pp. 242–256.
Search in Google Scholar Back to article
[4] HOFER, R.—LARCHER, G.: On existence and discrepancy of certain digital Niederreiter-Halton sequences, Acta Arith. 141 (2010), 369–394.10.4064/aa141-4-5
Search in Google Scholar Back to article
[5] KAJIURA, H.—MATSUMOTO, M.—SUZUKI, K.: Characterization of matrices B such that (I, B, B2) generates a digital net with t-value zero, Finite Fields Appl. 52 (2018), 289–300.10.1016/j.ffa.2018.04.011
Search in Google Scholar Back to article
[6] LOH, W. L.: On the asymptotic distribution of scrambled net quadrature, Ann. Statist. 31 (2003), 1282–1324.10.1214/aos/1059655914
Search in Google Scholar Back to article
[7] NIEDERREITER, H.: Point sets and sequences with small discrepancy, Monatsh. Math. 104 (1987), 273–337.10.1007/BF01294651
Search in Google Scholar Back to article
[8] ––––––Random Number Generation and Quasi-Monte Carlo Methods.In: CBMS-NSF Regional Conference Series, Applied Mathematics Vol. 63, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
Search in Google Scholar Back to article
[9] SOBOL’, I. M.: Distribution of points in a cube and approximate evaluation of integrals, Ž. Vyčisl. Mat. i Mat. Fiz. 7 (1967), 784–802. (In Russian)
Search in Google Scholar Back to article
[10] TEZUKA, S.: A Generalization of Faure Sequences and its Efficient Implementation, Research Report IBM RT0105 (1994), 1–10. Retrieved as a preprint 2nd of August 2018 from https://www.researchgate.net/publication/311808492_A_generalization_of_Faure_sequences_and_its_efficient_implementation
Search in Google Scholar Back to article

A Complete Classification of Digital (0,m, 3)-Nets and Digital (0, 2)-Sequences in Base 2

Abstract

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