## Niederreiter–Xing Sequence Construction III Using Non-Rational Places

In [1] Niederreiter and Xing develop a method (Niederreiter-Xing (NX) sequence construction III) for creating a digital (*t*, *s*)-sequence over **F**_{b} based on an algebraic function field of genus *g* containing *s* + 1 distinct places *P*_{∞}, *P*_{1},…, *P*_{s} with deg *P*_{∞} = 1. The *t*-parameter is bounded by

*t*≤

*g*+ (deg

*P*

_{i}− 1).

NX sequence III is constructive, assuming that defining equations for the function field are given and that appropriate places *P*_{∞}, *P*_{1},…, *P*_{s} are known.

Usually, the best results are achieved if only rational places are used. However, a number of function fields are known where better results can be obtained by including a small number of places of higher degree.

### See Also

(Part of) Construction 18 in [2].

### References

[1] | Chaoping Xing and Harald Niederreiter. A construction of low-discrepancy sequences using global function fields. Acta Arithmetica, 73(1):87–102, 1995.MR1358190 (96g:11096) |

[2] | Andrew T. Clayman, Kenneth Mark Lawrence, Gary L. Mullen, Harald Niederreiter, and Neil J. A. Sloane. Updated tables of parameters of ( t, m, s)-nets.Journal of Combinatorial Designs, 7(5):381–393, 1999.doi:10.1002/(SICI)1520-6610(1999)7:5<381::AID-JCD7>3.0.CO;2-S MR1702298 (2000d:05014) |

### 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–Xing Sequence Construction III Using Non-Rational Places.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_SNX3NonRationalPlaces.html