Niederreiter Sequence

The Niederreiter sequences [1] are digital (tb,s, s)-sequence over Fb, where b is a prime power and

tb,s = $\displaystyle \sum_{{i=1}}^{{s}}$(deg pi − 1),

with pi denoting the ith monic irreducible polynomial over Fb ordered by degree.

The Niederreiter sequence was the first known construction yielding (t, s)-sequences for arbitrary dimensions s and arbitrary prime powers b. For many pairs b, s it is still the sequence with the best t-parameter known today. Since it is fairly easy to implement, most computer programs generating (t, s)-sequences for arbitrary bases and dimensions are based on this construction.

For s = 1 the Niederreiter sequence (defined by the polynomial p1 = x) is a (0, 1)-sequence identical to the van der Corput sequence in the same base.

See Also

References

[1]Harald Niederreiter.
Low-discrepancy and low-dispersion sequences.
Journal of Number Theory, 30(1):51–70, September 1988.
doi:10.1016/0022-314X(88)90025-X MR960233 (89k:11064)
[2]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. “Niederreiter Sequence.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_SNiederreiter.html

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