(0, m, d)-Net Property: Optimal Equidistribution

• The (0, m, d)-net property is defined by every b-adic box at level m containing exactly one point, which guarantees optimal uniformity.
• Digital constructions employing generating matrices over finite fields enable precise, low-discrepancy point sets for error-bound numerical integration.
• Applications of (0, m, d)-nets include quasi-Monte Carlo methods, discrepancy theory, and randomized integration, with strong negative dependence under scrambling.

A $(0, m, d)$-net in base $b$ is a finite point set $P \subset [0,1)^d$ with cardinality $|P| = b^m$ such that every $b$-adic box (elementary interval) of total $b$-adic level $m$ contains exactly one point of $P$. These nets are the extremal case of $(t, m, d)$-nets with $t=0$—the optimal equidistribution property for digital nets. They play a central role in quasi-Monte Carlo methods, discrepancy theory, and randomized numerical integration, and have deep connections to the structure of digital nets, negative dependence under scrambling, explicit lattice constructions, and probabilistic existence in random sets.

1. Formal Definition and Properties

Let $b \ge 2$ (integer base), $d \ge 1$ (dimension), and $m \ge 0$ (integer parameter). For each multi-index $c = (c_1, \ldots, c_d) \in \mathbb{N}^d$ with $c_1 + \cdots + c_d = m$, and each $a = (a_1, \ldots, a_d)$ with $0 \le a_j < b^{c_j}$, define the elementary interval

$$E_{c,a} = \prod_{j=1}^d \left[\frac{a_j}{b^{c_j}}, \frac{a_j + 1}{b^{c_j}}\right) \subset [0, 1)^d$$

which partitions the unit cube into $b^m$ disjoint $b$-adic cells. A set $P \subset [0,1)^d$, $|P| = b^m$, is a $(0, m, d)$-net in base $b$ if and only if

$$|P \cap E_{c,a}| = 1 \quad \text{for every elementary interval } E_{c,a} \text{ with } c_1 + \cdots + c_d = m$$

equivalently, each $b$-adic box at level $m$ contains exactly one point of $P$ (Suzuki et al., 17 Dec 2025, Faure et al., 2014).

The property ensures maximal uniformity: every admissible $b$-adic partition at the prescribed scale is perfectly filled, preventing local concentration or voids.

2. Constructions and Equivalent Characterizations

Digital Construction

In the standard digital construction (for $b=p$ prime), fix $d$ matrices $C_1, \ldots, C_d \in \mathbb{F}_p^{m \times m}$. For every composition $\ell_1+\cdots+\ell_d = m$, the collection of the first $\ell_1$ rows of $C_1$, first $\ell_2$ of $C_2$, ..., first $\ell_d$ of $C_d$ forms a linearly independent set in $\mathbb{F}_p^m$. Enumerate $n=0,\ldots,p^m-1$, expand in base $p$, and form for each $j$: $x_n^{(j)} = \sum_{k=1}^m y_k^{(j)}(n)\,p^{-k},$ where $y^{(j)}(n) = C_j (n_0, \ldots, n_{m-1})^T$. The set $P = \{(x_n^{(1)},\ldots,x_n^{(d)})\}$ is a digital $(0, m, d)$-net in base $p$ (Faure et al., 2014).

Lattice and Polynomial Analogues

For prime $b$, consider an admissible $\mathbb{F}_b[x]$-lattice $X \subset \mathbb{F}_b((x^{-1}))^d$ generated by an invertible $T \in \mathrm{GL}_d(\mathbb{F}_b((x^{-1})))$. The node set is

$P_{f,d} = \{z \in X': z \in U_b^d\}$

with $$U_b^d = \{(f^{(1)},\ldots,f^{(d)}): \deg(f^{(j)})<0\}$$ and $X'$ suitably shrunken. After base-$b$ digit extraction, $\phi(P_{f,d})$ is a $(0, m, d)$-net whenever the minimal NRT- (Niederreiter–Rosenbloom–Tsfasman) weight on the dual group satisfies $\minNRT(P) \ge m+1$ (Dick et al., 2017).

Invariance

If every generating matrix $C_j$ is right-multiplied by an invertible $D \in \mathbb{F}_p^{m \times m}$, the point set is simply permuted—discrepancy and net properties are preserved (Faure et al., 2014).

3. Discrepancy and Quality Estimates

General Star-Discrepancy Bounds

For a $(0, m, d)$-net $P$ in base $b$,

$$D^*(P) \le C(d, b) (\log N)^{d-1}/N, \qquad N = b^m$$

with $C(d,b)$ explicit and independent of $m$ (Faure et al., 2014, Dick et al., 2017).

Dimension $d=2$:

• Niederreiter: $D^*(P)\le 2m+6$ (for $b=2$)
• Walsh bound (digital nets over $\mathbb{F}_2$): $D^*(P) \le \frac{m}{3} + \frac{19}{9}$
• Hammersley bound: $D^*(P) \le \frac{b-1}{2b} m + O(1)$
• Optimized permutations allow $D^*(P)/m$ to be reduced to $\frac{b-1}{8b}$ or $\frac{b^2}{8(b+1)}$ for specific choices (Faure et al., 2014).

Integration Error

Koksma–Hlawka yields, for any function $\psi$ of bounded Hardy–Krause variation,

$$|Q_P(\psi) - \int_{[0,1]^d} \psi(x)\,dx| \le D^*(P) \cdot \mathrm{Var}_{HK}(\psi)$$

so $(0,m,d)$-nets enable nearly optimal quadrature in terms of discrepancy (Dick et al., 2017).

Lower Bound

No $(0, m, d)$-net can have star-discrepancy below $c \log N$ with $c>0$ absolute; for $d=2$, $D^*\ge 0.03\, m\log b$ for $N=b^m$ (Faure et al., 2014).

4. Dependence Structure and Negative Dependence Under Scrambling

Given a digital $(0, m, d)$-net $P_n$, for each multi-index $\mathbf{k}$, define

$$C_b(\mathbf{k}; P_n) = \frac{b^{|\mathbf{k}|}\, M_b(\mathbf{k}; P_n)}{n(n-1)}, \text{ where } M_b(\mathbf{k}; P_n) = \#\{ (\mathbf{v},\mathbf{w})\in P_n\times P_n, \mathbf{v}\neq\mathbf{w}: \mathbf{v},\mathbf{w}\text{ in same } I_{\mathbf{k}}\}.$$

A scrambled digital $(0, m, d)$-net is negative lower orthant dependent (NLOD) if and only if $C_b(\mathbf{k}; P_n) \le 1$ for all $\mathbf{k}$ (Wiart et al., 2019). This property—termed "complete quasi-equidistribution"—is strictly stronger than $t=0$ alone and guarantees that the variance of quasi-monotone function estimators is never worse than plain Monte Carlo, for all sample sizes.

No digitally scrambled $(t, m, d)$-net with $t>0$ is NLOD; only $(0, m, d)$-nets enjoy this property. The full array $\{C_b(\mathbf{k};P_n)\}$ serves as a fine-grained quality metric, transcending the classical $t$ parameter (Wiart et al., 2019).

5. Probabilistic Existence and Pattern Counting

Let $S_N$ be a set of $N$ independent uniform points in $[0,1)^d$. Denote $C_{b,d}(N,m)$ as the event that $S_N$ contains a $(0, m, d)$-net as a subset. The number of geometric patterns that can support a $(0, m, d)$-net is

$a_{b,d}(m) \le (b!)^{m\,b^{m-1}(d-1)}$

with equality for $d=2$ via an explicit "strip-and-permute" induction (Suzuki et al., 17 Dec 2025).

Probability bounds depend on occupancy arguments: $P(C_{b,d}(N,m)) \le a_{b,d}(m) \cdot p_N(b^m)$ where $p_N(b^m)$ is the probability that $b^m$ specific cubes each contain at least one point. Explicit exponential bounds and Paley–Zygmund inequalities reveal that, for $m,N\to\infty$:

• If $N \gg b^{md} m \log b$, then $P(C_{b,d}(N, m)) \to 1$: random point clouds of this size "hide" a $(0, m, d)$-net with high probability.
• If $N \ll b^{md} (b!)^{m(d-1)/b}$, then $P(C_{b,d}(N, m)) \to 0$: not enough points to ensure coverage (Suzuki et al., 17 Dec 2025).

This probabilistic threshold refines the understanding of random quasi-Monte Carlo and the hidden regularity within high-volume random clouds.

6. Applications and Impact in Numerical Integration

$(0, m, d)$-nets underlie the most effective quasi-Monte Carlo cubature rules for cubical domains, as their low discrepancy yields precise error bounds for functions of bounded Hardy–Krause variation. They feature in digital net methods, high-dimensional integration, and serve as templates for randomized QMC algorithms.

Polynomial analogues (in $\mathbb{F}_b[x]$) have enabled the construction of nets with the $t=0$ property for dimensions up to $d=b^n$ (for prime $b$), particularly with admissible Vandermonde-type lattices, achieving optimality in star-discrepancy bounds and bridging combinatorial and analytic aspects of QMC (Dick et al., 2017). The insight that only scrambled $(0, m, d)$-nets attain full negative dependence substantiates their special importance in randomized algorithms (Wiart et al., 2019).

7. Structural Remarks and Low-Dimensional Peculiarities

• In $d=1$, the concept is trivial, as any $N$-point set can be ordered to match any prescribed digit pattern.
• In $d=2$, the Hammersley net with identity permutation is the unique $(0, m, 2)$-net constructed via standard digital methods; digital permutations allow reduction of discrepancy constants, though new lower bounds prohibit further improvement beyond certain thresholds (Faure et al., 2014).
• Nontrivial $(0, m, d)$-nets exist for $d \le b$ via explicit polynomial methods, with generalizations for $d=b^k$ and specially structured lattices.

The $(0, m, d)$-net property establishes a foundation for both explicit construction and stochastic emergence of highly regular, low-discrepancy point sets, shaping the modern understanding and efficacy of deterministic and randomized quasi-Monte Carlo theory.