Figures of merit are the quality measures used to evaluate point sets.

The lower the merit value, the better the point set as the integration error will be smaller. Figures of merit are often derived from Koskma-Hlawka type inequalities which bound the integration error.

LatNet Builder implements various figures of merit. Figures of merit are specific to a construction method. Most of them can be parametrized by a norm-type and weights.

Let \(P_n = \left\{(x_{0,1}, \dots, x_{0, s}), \dots, (x_{n-1,1}, \dots, x_{n-1, s}) \right\}\) denote a point set in dimension \(s\) with \(n\) points, \( q \leq +\infty \) a norm-type and \(\gamma_{\mathfrak u}\) some general weights.

Figures of merit implemented by LatNet Builder have the following form:

\[ D(P_n)^q = \sum_{\emptyset \neq \mathfrak u \subseteq \{1, \dots, s\}}\left(\gamma_{\mathfrak u} D_{\mathfrak u}(P_n)\right)^q, \]

where \(D(P_n)\) is the merit value of the point set \(P_n\) and \( D_{\mathfrak u}(P_n) \) its projection-dependent merit value. If \(q = +\infty\), the sum in the above equation is replaced by a maximum and the exponent \(q\) taken equal to 1. Note that the figure of merit actually computed by LatNet Builder is not \(D(P_n)\) but rather \(D(P_n)^q\).

LatNet Builder implements various projection-dependent merits depending on the point set type: