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 φb : ℕ0→[0, 1). If n ∈ ℕ0 has the b-adic expansion

n = $\displaystyle \sum_{{j=1}}^{{T}}$ajbj−1

with aj ∈ {0,…, b – 1}, then φb is defined as

φb(n) := $\displaystyle \sum_{{j=1}}^{{T}}$aj/bj.

In other words, the jth b-adic digit of n becomes the jth b-adic digit of φb(n) behind the decimal point. The van der Corput sequence in base b is defined as (φb(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: 2024-09-05. http://mint.sbg.ac.at/desc_SCorput.html

Show usage of this method