off-topic connection between Definition \[def:ConsGen\] and combinatorial tilings. If in addition to being surjective, the reduction $\phi_p$ is a bijection when restricted to a set $\mathcal{P} \subset \Lambda$ of cardinality $p^n$, then $\mathcal{P}$ tiles $\Lambda$ by translations of vectors of $\Lambda_p$.

This framework contains classical Construction A [@Splag], [@Rush1989], the constructions in [@KosiOngOggier], and [@ConsAOggier]. We derive sufficient conditions for this general construction to admit a Minkowski-Hlawka theorem. Set $\beta= {V^{1/m}}/{(p^{n-k}V(\Lambda)^{1/m})}$ and let $$\mathbb{L}_p=\left\{ \beta \Lambda_p(C) : C \mbox{ is an } (n,k,p)-\mbox{code}\right\}
\label{eq:ensemble}$$ be the ensemble of all lattices associated to codes of dimension $k$, normalized to volume $V$. Suppose that a lattice in $\mathbb{L}_p$ is picked at random by choosing $C$ uniformly. We shall prove a generalized version of the Minkowski-Hlawka theorem for $\mathbb{L}_p$. Instead of functions with bounded support, we will consider a wider class of functions. Let $W=\mbox{span}(\Lambda)$ be the minimal subspace of $\mathbb{R}^n$ containing $\Lambda$ (therefore, $\mbox{dim}( \mbox{span}(\Lambda_p(C))) = m$).

Let $f:W\to \mathbb{R}$ be a Riemann-integrable function. We say that $f$ is *semi-admissible* if $$|f({\mathbf{x}})| \leq \frac{b}{(1+\left\| {\mathbf{x}}\right\|)^{m+\delta}}, \forall {\mathbf{x}}\in W$$ where $b > 0$ and $\delta > 0$ are positive constants.

Notice that any bounded integrable function with compact support is semi-admissible. Of particular 