method of Rogers [@Rogers], choosing $f(x)$ appropriately in the Minkowski-Hlawka theorem. We shall give a complete proof in the next section, in the context of $\mathcal{O}_K$-lattices.

Constructions From Number Fields {#sec:numbFields}
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From now on we consider constructions of random ensembles based on algebraic number theory. We refer the reader to [@Fermat] and [@Notes] for an introduction to the theory, as well as undefined notation.

Let $K/\mathbb{Q}$ be a number field with degree $n$ and signature $(r_1,r_2)$. Denote its real embeddings by $\sigma_1,\ldots,\sigma_{r_1}$ and their pairs of complex embeddings by $$\sigma_{r_1+1}, \overline{\sigma_{r_1+1}},\ldots,\sigma_{r_1+r_2+1},\overline{\sigma_{r_1+r_2+1}}.$$ Let $\mathcal{O}_K$ be the ring of integers of $K$ and $\mathcal{I}\subset \mathcal{O}_K$ be an ideal. An ideal can be identified with a real lattice of dimension $(r_1+2r_2)$ via the canonical embedding $$\sigma : \mathcal{O}_K \to \mathbb{R}^{r_1+2r_2}$$ $$\begin{split}\sigma(x) = (\sigma_1(x),\ldots,\sigma_{r_1}(x),&\Re \sigma_{r_1+1}(x),\ldots \Re \sigma_{r_1+r_2+1}(x),\\ &\Im \sigma_{r_1+1}(x),\ldots,\Im \sigma_{r_1+r_2+1}(x)).
\end{split}$$ Lattices constructed from the embedding of ideals $\mathcal{I} \subset \mathcal{O}_K$ are called *ideal lattices*, and appear in the study of modular forms, coding theory, and cryptography. In this section we study the Minkowski Hlawka theorem for $\mathcal{O}_K$-lattices and related structures.

Let $E=K\otimes_{\mathbb{Q}} \mathbb{R} $ be the Euclidean space generated by $K$. An $\mathcal{O}_K$-lattice is a free $\mathcal{O}_K$ sub-module of $E^t$, for some 