Generalized Faure Sequence

The Faure sequences are a digital (0, s)-sequence over Fb with b denoting a prime (original case) or a prime power (general case) greater or equal to s. The case for b prime was proven by Faure [1], the general result is due to Niederreiter [2, Theorem 6.18].

The s infinite generator matrices C(1),…,C(s) over Fb are defined by C(i) = (cjr(i))j, r ≥ 0 with

cjr(i) = $\displaystyle \binom{r}{j}$αir−j,

where α1,…, αs denote s distinct elements from Fb and the conventions α0 = 1 for all α ∈ Fb and $ \binom{r}{j}$=0 for j > r are used.

For α = 1, the resulting matrix is the infinite Pascal matrix modulo the characteristic of Fb; for α = 0, it is the infinite identity matrix. If s = 1 and α1 = 0, the resulting (0, 1)-sequence is identical to the van der Corput sequence in the same base.

Sequences with the same parameters can also be obtained using Niederreiter sequences or Niederreiter-Xing sequences with rational function fields.

See Also

References

[1]Henri Faure.
Discrépance de suites associées à un système de numération (en dimension s).
Acta Arithmetica, 41:337–351, 1982.
[2]Harald Niederreiter.
Point sets and sequences with small discrepancy.
Monatshefte für Mathematik, 104(4):273–337, December 1987.
doi:10.1007/BF01294651 MR918037 (89c:11120)
[3]Gary L. Mullen, Arijit Mahalanabis, and Harald Niederreiter.
Tables of (t, m, s)-net and (t, s)-sequence parameters.
In Harald Niederreiter and Peter Jau-Shyong Shiue, editors, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, volume 106 of Lecture Notes in Statistics, pages 58–86. Springer-Verlag, 1995.

Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.
Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Generalized Faure Sequence.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_SFaure.html

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