g}}_i: \ t_i\in {\ensuremath{\mathbb{Z}}}\right\}.$$ Lattice bases are not unique. A point ${\ensuremath{\mathbf g}}\in L\setminus \{{\ensuremath{\mathbf 0}}\}$ is called a *primitive point* of $L$ if the line segment joining the origin ${\ensuremath{\mathbf 0}}$ and ${\ensuremath{\mathbf g}}$ does not contain any other point of $L$.

The *dual lattice* $L^\bot$ of $L$ is defined as $$L^\bot = \left\{{\ensuremath{\mathbf z}}\in {\ensuremath{\mathbb{R}}}^s:\ {\ensuremath{\mathbf z}}\cdot {\ensuremath{\mathbf x}}\in {\ensuremath{\mathbb{Z}}}, \text{ for all } {\ensuremath{\mathbf x}}\in L  \right\},$$ where ${\ensuremath{\mathbf z}}\cdot {\ensuremath{\mathbf x}}$ denotes the usual inner product.

An $s$-dimensional *integration lattice* is an $s$-dimensional lattice that contains ${\ensuremath{\mathbb{Z}}}^s$ as a sublattice. An $s$-dimensional $N$-point *lattice rule* is an $s$-dimensional integration lattice $L$ of the form $$\label{eqn:integrationlattice}
L = \bigcup_{n=0}^{N-1} \left( {\ensuremath{\mathbf x}}_n + {\ensuremath{\mathbb{Z}}}^s  \right),$$ where ${\ensuremath{\mathbf x}}_0, \ldots, {\ensuremath{\mathbf x}}_{N-1}$ are the $N$ distinct points of $L$ that belong to $[0,1)^s$.

The spectral test for lattices is defined as follows.

\[definition:classicalspectraltest\] Let $L$ be an $s$-dimensional lattice in $\mathbb{R}^s$. We will call a family ${\mathcal{H}_ C}$ of parallel hyperplanes ${H_{c}}$, $c \in C$, in $\mathbb{R}^s$ *a cover of $L$* if (i) $L \subseteq \bigcup_{c\in C} {H_{c}}$, and (ii) $C$ is the smallest set (in the sense of set-inclusion) with this property.

Let ${\mathcal{H}_ C}$ denote an 