## Niederreiter Sequence

The Niederreiter sequences [1] are digital (*t*_{b,s}, *s*)-sequence over **F**_{b}, where *b* is a prime power and

*t*

_{b,s}= (deg

*p*

_{i}− 1),

with *p*_{i} denoting the *i*th monic irreducible polynomial over **F**_{b} 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 *p*_{1} = *x*) is a (0, 1)-sequence identical to the van der Corput sequence in the same base.

### See Also

Construction 9 in [2].

### 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: 2008-04-04.
http://mint.sbg.ac.at/desc_SNiederreiter.html