## Van der Corput Sequence

The van der Corput sequence [1] is a digital (0, 1)-sequence, which exists for all bases *b* ≥ 2. It is defined by the radical inverse function *φ*_{b} : ℕ_{0}→[0, 1). If *n* ∈ ℕ_{0} has the *b*-adic expansion

*n*=

*a*

_{j}

*b*

^{j−1}

with *a*_{j} ∈ {0,…, *b* – 1}, then *φ*_{b} is defined as

*φ*

_{b}(

*n*) :=

*a*

_{j}/

*b*

^{j}.

In other words, the *j*th *b*-adic digit of *n* becomes the *j*th *b*-adic digit of *φ*_{b}(*n*) behind the decimal point. The van der Corput sequence in base *b* is defined as (*φ*_{b}(*n*))_{n ≥ 0}.

A generator matrix (over an arbitrary commutative ring with identity and cardinality *b*) is the infinite identity matrix.

A (0, *m*, 2)-net derived from the van der Corput sequence using the propagation rule net from sequence is usually called Hammersley net.

### References

[1] | Johannes Gualtherus van der Corput. Verteilungsfunktionen. Proc. Ned. Akad. v. Wet., 38:813–821, 1935. |

### 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. “Van der Corput Sequence.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_SCorput.html