norm, i.e. $nrd(x) = \det \phi(x) \text{ (mod }p)$, for any $x \in \mathcal{H}$.

Algorithmic Effectiveness {#sec:conc}
=========================

Theorem \[thm:2\] holds in the limit $p_j \to \infty$. However, for each $n$, under some conditions it is possible to find *finite* ensembles that contain dense lattices. In the literature, this is referred to as *effectiveness* (e.g. [@Splag p. 18] and [@Zemor]). We show conditions for a family of reductions to be effective. We need the following lemma, which is a special case of a classical result in the Geometry of Numbers (see [@Leker p. 141]) and is also valid if $\mathcal{B}_r$ is replaced by more general sets. We include a proof here for the sake of completeness.

Let $\mathcal{P}$ be a fundamental region for $\Lambda$, an let $l_0 = \sup_{{\mathbf{x}}\in\mathcal{P} }\left\|   {\mathbf{x}}\right\|.$ For $r > l_0$, we have $$(r-l_0)^n V_n \leq V(\Lambda) {\mathcal{N}_{\Lambda}(r)} \leq (r+l_0)^nV_n.$$ In particular, we can take $l_0 = \tau(\Lambda)$ to be the covering radius of $\Lambda$. \[lem:pointsEnumerator\]

We show the set inclusion $$\mathcal{B}_{r-l_0} \subset \bigcup_{{\mathbf{x}}\in\Lambda\cap \mathcal{B}_{r}}({\mathbf{x}}+\mathcal{P})\subset \mathcal{B}_{r+l_0}.$$ The lemma then follows from a simple volume calculation of the three sets.

If ${\mathbf{y}}= {\mathbf{x}}+ \mathbf{p}$, ${\mathbf{x}}\in \Lambda\cap\mathcal{B}_r$, $\mathbf{p} \in \mathcal{P}$, then $\left\| {\mathbf{y}}\right\| \leq \left\| {\mathbf{x}}\right\|+\left\| \mathbf{p}\right\|  \leq 