Digital point-set and rank-one lattice cube quadrature

Inputs: a dimension $d\ge 1$, a function $f:[0,1{]}^d\to \mathbb{R}$ that is finite at each generated node, and exactly one of the following point-set specifications:

1. Digital branch: a prime $p$, an integer $m\ge 1$, and matrices $C_1,\dots ,C_d\in {\mathbb{F}}_p^{m\times m}$. Optionally supply an integer $t$ with $0\le t\le m$ when the caller also wants the exact $(t,m,d)$-net property checked.
2. General lattice-rule branch: a full-rank discrete additive subgroup $L\subset {\mathbb{R}}^d$ containing ${\mathbb{Z}}^d$, with finite quotient $L\mathrm{/}{\mathbb{Z}}^d$, and a supplied complete list $P$ of the distinct coset representatives in $[0,1{)}^d$.
3. Rank-one lattice branch: an integer $N\ge 1$ and $z=(z_1,\dots ,z_d)\in {\mathbb{Z}}^d$ with $\gcd (z_1,N)=1$, where $\gcd$ has the convention of Common divisor, and the greatest common divisor $\gcd (a,b)$, with the convention $\gcd (0,0):=0$.

Outputs: The ordered point list $P$, the finite node average $A_P(f)$, and, when $t$ is supplied in the digital branch, the net-count check result.

A lattice rule means the equal-weight rule on the representatives of an integration lattice $L$ as in branch 2; its node count is $N=\mathrm{\mid }L\mathrm{/}{\mathbb{Z}}^d\mathrm{\mid }$. Branch 3 specifies the rank-one fractional-part construction.

Instructions:

1. In the digital branch, for each $n=0,\dots ,p^m-1$, write $n={\sum }_{r=1}^m{n}_{r-1}{p}^{r-1}$, form $v(n)=(n_0,\dots ,n_{m-1}{)}^T\in {\mathbb{F}}_p^m$, and compute $y^{(j)}=C_{j}v(n)$ in ${\mathbb{F}}_p^m$. Replace each coordinate $y_r^{(j)}$ by its representative in $\{0,\dots ,p-1\}$ and set $x_{n,j}={\sum }_{r=1}^m{y}_r^{(j)}{p}^{-r}$. The output point list is $P=(x_0,\dots ,x_{p^m-1})$. If $t$ was supplied, enumerate every integer tuple $k_1,\dots ,k_d\ge 0$ with ${\sum }_j{k}_j=m-t$ and every tuple $a_j\in \{0,\dots ,p^{k_j}-1\}$; count the output nodes in each corresponding elementary interval. Return “net check passed” exactly when every count equals $p^t$, and otherwise return “not a $(t,m,d)$-net.”
2. In the general lattice-rule branch, take the supplied complete list $P$ of representatives; do not add further points or repeat any coset representative.
3. In the rank-one lattice branch, for each $n=0,\dots ,N-1$, set $x_{n,j}=\{n{z}_j\mathrm{/}N\}$ for $j=1,\dots ,d$, where $\{a\}:=a-\lfloor a\rfloor \in [0,1)$ and $\lfloor a\rfloor$ is the integer part from Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$. The output point list is $P=(x_0,\dots ,x_{N-1})$.
4. In each branch, let $N$ be the output list length and return the finite node average $A_P(f):=N^{-1}{\sum }_{n=0}^{N-1}f(x_n)$ together with $P$. In the digital branch $N=p^m$; in the general lattice branch $N=\mathrm{\mid }L\mathrm{/}{\mathbb{Z}}^d\mathrm{\mid }$. When $f$ is bounded and Riemann integrable, $A_P(f)=Q_P(f)$ and its target is the cube integral $I_d(f)$ of Star discrepancy, digital nets, and digital sequences. For other admissible inputs the finite average is still defined, but no cube-integral target is asserted. The algorithm does not estimate a discrepancy, a variation, or a sampling variance, and a generated digital point set has no claimed net quality unless the optional count check passes.