## Faure Sequence

The Faure sequences are a digital (0, *s*)-sequence over **F**_{b} 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)},â€¦,**C**^{(s)} over **F**_{b} are defined by **C**^{(i)} = (*c*_{jr}^{(i)})_{j, r â‰¥ 0} with

*c*

_{jr}

^{(i)}=

*Î±*

_{i}

^{râˆ’j},

where *Î±*_{1},â€¦, *Î±*_{s} denote *s* distinct elements from **F**_{b} and the conventions *Î±*^{0} = 1 for all *Î±* âˆˆ **F**_{b} and =0 for *j* > *r* are used.

For *Î±* = 1, the resulting matrix is the infinite Pascal matrix modulo the characteristic of **F**_{b}; for *Î±* = 0, it is the infinite identity matrix. If *s* = 1 and *Î±*_{1} = 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

ConstructionÂ 8 in [3].

### References

[1] | Henri Faure. DiscrÃ©pance de suites associÃ©es Ã un systÃ¨me de numÃ©ration (en dimension s).Acta Arithmetica, 41:337â€“351, 1982. |

[2] | Harald Niederreiter. Point sets and sequences with small discrepancy. Monatshefte fÃ¼r Mathematik, 104(4):273â€“337, December 1987.doi:10.1007/BF01294651 MR918037 (89c:11120) |

[3] | GaryÂ 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 58â€“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. “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-ori.html