This page describes how interlaced digital nets and polynomial lattice rules can be constructed using the command-line tool.

LatNet Builder supports the construction of interlaced digital nets and interlaced polynomial lattice rules. More precisely, it constructs the underlying digital net of the interlaced digital net, using various quality criteria.

The available figures of merit are the \(B_{\alpha, d, (1)}\) and the \(B_{d, (2)}\) interlaced discrepancies from [10]. The interlacing factor (–interlacing-factor) must be greater than one to enable these figures of merit. They are only available with an \(\ell_1\) norm. This allows a coordinate-uniform evaluation algorithm which should always be used when searching for good interlaced digital nets.

The output of LatNet Builder is the underlying digital net (or polynomial lattice rule) of the interlaced digital net (or interlaced polynomial lattice rule). The digit interlacing function should be applied to this point set in dimension \(ds\) to obtain the interlaced point set in dimension \(s\).

Constructing interlaced polynomial lattice rules

As for polynomial lattice rules, interlaced polynomial lattice rules are available on both side of the software, through a lattice implementation and a digital net implementation:

Lattice implementation (–set-type lattice):
- the fast-CBC algorithm is only available in this implementation. Because of the restriction on the norm-type, the coordinate-uniform evaluation algorithm is always enabled. Thus, it is always possible to use the fast-CBC algorithm for non-random CBC searches,
- normalizations are only supported by this implementation. They are available for both quality criteria,
- embedded interlaced polynomial lattice rules are only available in the lattice implementation;
Net implementation (–set-type net):
- this implementation should only be used for exhaustive, random or random-CBC explorations.