A multilevel point set is a finite sequence of nested point sets \(\{P_{n_m}, 1 \leq m \leq k \}\): for each embedding levels \(l \leq m\), \(P_{n_l} \subseteq P_{n_m}\).

Constructing multilevel point sets involves producing a compound merit value that accounts for the merit of each nested level. This compound merit value is computed as follows:

the figure of merit is evaluated for each nested level: the merit value of the embedding level \(m\) is denoted by \(D_q(P_{n_m})\);
optionally, multilevel filters and normalizations are applied to the individual merit values of each nested level;
these filtered individual merit values are combined using a sum or a maximum into a single value:
\[ \bar{D}(P_{n_1}, \dots, P_{n_k})^q = \bigotimes_{m = 1, \dots, k} D(P_{n_m})^q. \]

The possible combiners are the sum and the max of the merit values.

Multilevel point sets are handled differently for lattices and for nets.

Embedded lattices

Embedded lattices [12] with \(m\) levels consist in taking a sequence of modulus \(p_m = b^{m}\) for \(1 \leq m \leq k\) with \(b\) a prime base for ordinary lattice rules or an irreducible polynomial for polynomial lattice rules. Then for each embedding level, we consider the associated lattice rule. The same generating vector is used for all the embedding levels. The sequence of lattice rules forms a multilevel point set.

Multilevel digital nets

For multilevel digital nets, we consider only generating matrices which are upper triangular with ones on the main diagonal. That way, the first \(2^m\) points of the corresponding digital net form a digital net whose generating matrices are the upper-left submatrices of size \(m \times m\) of the original generating matrices.

Multilevel digital nets are only available with the Sobol' and explicit constructions. In the case of the Sobol' construction, the generating matrices are upper triangular with ones on the main diagonal. For the explicit construction, we restrict the search space to upper triangular matrices with ones on the main diagonal.