Base Change

Let u and uʹ be two integers with gcd(u, uʹ) = 1 and m a multiple of uʹ. In [1] (originally in [2, Section 4.2]) it is shown that every (t, m, s)-net in base bu is a (tʹ, mʹ, s)-net in base buʹ with mʹ = mu/uʹ and

tʹ ≤ min{$\displaystyle \left\lceil\vphantom{ \frac{ut+(s−1)(u−1)}{uʹ}}\right.$$\displaystyle {\frac{{ut+(s−1)(u−1)}}{{uʹ}}}$$\displaystyle \left.\vphantom{ \frac{ut+(s−1)(u−1)}{uʹ}}\right\rceil$,$\displaystyle \left\lceil\vphantom{ \frac{ut+mʹ(-uʹ\bmod u)}{uʹ+(-uʹ\bmod u)}}\right.$$\displaystyle {\frac{{ut+mʹ(-uʹ\bmod u)}}{{uʹ+(-uʹ\bmod u)}}}$$\displaystyle \left.\vphantom{ \frac{ut+mʹ(-uʹ\bmod u)}{uʹ+(-uʹ\bmod u)}}\right\rceil$}.

For u = 1 this formula reduces to that of base expansion, for uʹ = 1 to that of base reduction. However, if u > 1 and uʹ > 2, the bound for tʹ is often better than applying base reduction from base bu to base b followed by base expansion from base b to base buʹ.

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: 2024-09-05. http://mint.sbg.ac.at/desc_NBChange.html

Show usage of this method