## Base Change

Let *u* and *u*ʹ be two integers with gcd(*u*, *u*ʹ) = 1 and *m* a multiple of *u*ʹ. In [1] (originally in [2, Section 4.2]) it is shown that every (*t*, *m*, *s*)-net in base *b*^{u} is a (*t*ʹ, *m*ʹ, *s*)-net in base *b*^{uʹ} with *m*ʹ = *mu*/*u*ʹ and

*t*ʹ ≤ min{,}.

For *u* = 1 this formula reduces to that of base expansion, for *u*ʹ = 1 to that of base reduction. However, if *u* > 1 and *u*ʹ > 2, the bound for *t*ʹ is often better than applying base reduction from base *b*^{u} to base *b* followed by base expansion from base *b* to base *b*^{uʹ}.

### References

[1] | Gottlieb Pirsic. Base changes for ( t, m, s)-nets and related sequences.Sitzungsberichte Österr. Akad. Wiss. Math.-Maturw. Kl. Abt. II, 208:115–122, 1999. |

[2] | Gottlieb Pirsic.Embedding Theorems and Numerical Integration of Walsh Series over Groups.PhD thesis, University of Salzburg, Austria, 1997. |

### 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. “Base Change.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_NBChange.html