## Sobol Sequence

The Sobolʹ sequences [1] are a digital (*t*_{s}, *s*)-sequences over ℤ_{2}, where

*t*

_{s}= (deg

*p*

_{i}− 1),

with *p*_{1} = *x* ∈ ℤ_{2}[*x*] and *p*_{i+1} denoting the *i*th primitive polynomial over ℤ_{2} ordered by degree.

Sobolʹ sequences were the first known constructions yielding (*t*, *s*)-sequences for arbitrary dimensions *s*. They were introduced long before the theory of (*t*, *s*)-sequences over arbitrary finite fields **F**_{b} was established in [2]. However, they only exist for *b* = 2, and even in this case, the resulting *t* parameter is not optimal for *s* > 3. For *s* > 7, even the Niederreiter sequence, which is equally easy to implement, yields lower *t*-values.

For *s* = 1 the Sobolʹ sequence (defined by the polynomial *p*_{1} = *x*) is a (0, 1)-sequence identical to the van der Corput sequence in base 2.

### See Also

Construction 6 in [3].

### References

[1] | Ilʹya M. Sobolʹ. On the distribution of points in a cube and the approximate evaluation of integrals. U.S.S.R. Computational Mathematics and Mathematical Physics, 7(4):86–112, 1967. |

[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. “Sobol Sequence.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_SSobol.html