LatNet Builder supports the following figures of merit for digital nets:

The \(\mathcal P_{\alpha}\) discrepancy [21] with \(\alpha \in \{2, 4, 6, 8\}\), using a coordinate-uniform evaluation algorithm with an \(\ell_2\) norm:
\[ \mathcal{D}^2_{\mathfrak u}(P_n) = \frac{1}{{n}} \sum_{i = 0}^{n-1} \prod_{j \in \mathfrak u} \omega_{\alpha}(x_{i,j}), \]
where
\[ \omega_{\alpha}(x) = \mu(\alpha) - 2^{(1+\lfloor \log_2(x) \rfloor) (\alpha - 1)}(\mu(\alpha) + 1), \]
where \( \mu(\alpha) = (1 - 2^{1- \alpha})^{-1} \) for even \( \alpha \).
The \(\mathcal R\) criterion [25], using a coordinate-uniform evaluation algorithm with an \(\ell_2\) norm:
\[ \mathcal{D}^2_{\mathfrak u}(P_n) = \frac{1}{{n}} \sum_{i = 0}^{n-1} \prod_{j \in \mathfrak u} \phi'_m(x_{i,j}), \]
where \(m\) is the degree of the modulus and
\[ \phi'_m(x) = \begin{cases} \frac{1}{2}i_0 & \textrm{ if } x = \sum_{i= i_0}^{\infty} \xi_i 2^{-i}, 1 \leq i_0 \leq m, \xi_{i_0} \neq 0 \\ 1 + \frac{1}{2}m & \textrm{otherwise.} \end{cases} \]

LatNet Builder also supports figures of merit based on the equidistribution of points. Take a digital net \(P_n \) in dimension \(s\) with \(n = 2^k\) points. Suppose we divide axis \(j\) in \(2^{q_j}\) equal parts for some integer \(q_j\), for each \(j\). This determines a partition of \([0,1)^s\) into \(2^{q_1+\dots+q_s}\) rectangles of equal sizes. If each rectangle contains exactly the same number of points, we say that the net is \((q_1,\dots,q_s)\)-equidistributed.

The following figures of merit are supported by LatNet Builder:

The t-value [6] merit, using the fast algorithm from [5], with weights equal to \(\gamma_{\mathfrak u} = \mathbb{1}_{\mathfrak u = \{1, \dots, s\}} \) and an \(\ell_{\infty}\) norm:
\(\mathcal{D}_{\mathfrak u}(P_n)\) is the t-value of the projection of \(P_n \) on \(\mathfrak u\). The sum which defines this figure has only one term and this term equals the t-value of the net. The choice of weights or norm-type has no impact on the ranking of digital nets according to this figure. The t-value of a net with \(n = 2^k\) points is the smallest \(t\) such that the net is \((q_1,\dots,q_s)\)-equidistributed whenever \(q_1+\dots+q_s = k-t \). This figure is not available with CBC exploration methods as it cannot be computed in a CBC-way.
The projection-dependent t-value [14] using state-of-the art algorithms, with an \(\ell_q\) norm:
\(\mathcal{D}_{\mathfrak u}(P_n)\) is the t-value of the projection of \(P_n \) on \(\mathfrak u\). Contrary to the t-value merit, this figure of merit does not only consider the t-value of the whole net but also the t-values of its subprojections.
The projection-dependent t-value-based star discrepancy bound (Corollary 5.3 in [6]), using state-of-the art algorithms, with an \(\ell_q\) norm:
\[ \mathcal{D}_{\mathfrak u}(P_n) = 2^{t_{\mathfrak u}(P_n)-k} \sum_{i=0}^{|\mathfrak{u}|-1} \binom{k-t_{\mathfrak u}(P_n)}{i} \]
where \(t_{\mathfrak u}(P_n)\) is the t-value of the projection of \(P_n \) on \(\mathfrak u\).
The projection-dependent resolution-gap [20], with an \(\ell_q\) norm:
\(\mathcal{D}_{\mathfrak u}(P_n)\) is the resolution-gap of the projection of \(P_n \) on \(\mathfrak u\). The resolution of a net is the biggest \(r\) such that the net is \((r,\dots,r)\)-equidistributed. The resolution is always smaller than \(\lfloor k/s\rfloor\). The resolution-gap is the difference between this upper-bound and the actual resolution.

Figures of merit for interlaced digital nets

Figures of merit specific to interlaced digital nets [10] are also supported by LatNet Builder. These figures of merit are the same as those used for interlaced polynomial lattice rules. These figures of merit for interlaced digital nets in dimension \(s\) can be seen as figures of merit for the underlying lattice rules in higher dimension \(ds\), where \(d\) is the interlacing factor. If \(P_n\) is the interlaced lattice rule, let \(\bar{P}_n = \left\{(z_{0,1}, \dots, z_{0, ds}), \dots, (z_{n-1,1}, \dots, z_{n-1, ds}) \right\}\) denote the underlying digital net in dimension \(ds\). This point of view also requires to transform (or interlace) weights in dimension \(s\) into weights in dimension \(d s\). Let \( \bar{\gamma} \) denote these interlaced weights.

The figures of merit for interlaced digital nets supported by LatNet Builder have the following form:

\[ D(P_n) = \sum_{\emptyset \neq \mathfrak u \subseteq \{1, \dots, d s\}}\left(\bar{\gamma}_{\mathfrak u} D_{\mathfrak u}(\bar{P}_n)\right), \]

The \(B_{\alpha, d, (1)}\) interlaced discrepancy, using an \(\ell_1\) norm with a coordinate-uniform evaluation algorithm:

\[ D_{\mathfrak u}(\bar{P}_n) = \sum_{i = 0}^{n-1} \prod_{j \in \mathfrak u} \phi_{\alpha, d, (1)}(z_{i, j}), \]

where \( \phi_{\alpha, d, (1)}(z) \) is defined by:
\[ \phi_{\alpha, d, (1)}(z) = \frac{1 - 2^{(\min(\alpha, d) -1) \lfloor \log_2(z) \rfloor} (2^{\min(\alpha, d)} -1)}{2^{(\alpha+2)/2} (2^{\min(\alpha, d) - 1} -1) }. \]

For this discrepancy, the interlaced weights are defined as:
\[ \bar{\gamma}_{\mathfrak u} = \gamma_{w(\mathfrak u)} 2^{\alpha (2d-1) |w(\mathfrak u)| / 2}, \]

where \(w\) denotes the weights interlacing operator:
\[ w(\mathfrak u) = \left\{ \left \lceil \frac{j}{d} \right\rceil, j \in \mathfrak u \right\}. \]
The \(B_{d, (2)}\) interlaced discrepancy, using an \(\ell_1\) norm with a coordinate-uniform evaluation algorithm:

\[ D_{\mathfrak u}(\bar{P}_n) = \sum_{i = 0}^{n-1} \prod_{j \in \mathfrak u} \phi_{d, (2)}(z_{i, j}), \]

where \( \phi_{d, (2)}(z) \) is defined by:
\[ \phi_{d, (2)}(z) = \frac{2^{d-1}(1 - 2^{(d -1) \lfloor \log_2(z) \rfloor} (2^{d} -1))}{(2^{d - 1} -1) }. \]

For this discrepancy, the interlaced weights are defined as:
\[ \bar{\gamma}_{\mathfrak u} = \gamma_{w(\mathfrak u)} \prod_{j \in \mathfrak u} 2^{- ( (j-1) \mod d) + 1}, \]

where \(w\) denotes the weights interlacing operator.