$\mathbb{F}_p^n$. For a general reduction we prove the following:

Let $\phi_{p}: \Lambda \to \mathbb{F}_{p}^n$ be a family of reductions (surjective homomorphisms), where $\Lambda$ is a lattice of rank $m$ in the Euclidean space. Consider the ensemble $$\mathbb{L}_{p} = \left\{\beta \phi_p^{-1}(\mathcal{C}): \mathcal{C} \mbox{ is a } k-\mbox{dimensional code in } \mathbb{F}_p^n\right\},$$ for an appropriate normalization factor $\beta$ so that all lattices have volume $V$. Denote by $\mathcal{N}_{\Lambda^{\prime}}(r) = \#( \mathcal{B}_r \cap \Lambda')$ the number of primitive points of $\Lambda$ inside a ball of radius $r$. If the first minimum of $\Lambda_{p} = \ker \phi_{p}$ satisfies $$\liminf_{p\to \infty}\left( \frac{\lambda_1(\Lambda_{p})}{p^{n/m}}\right) > 0, \mbox{ then }
    \label{eq:nonDeg}$$ $$\lim_{p\to\infty} E_{\mathbb{L}_{p}}[\mathcal{N}_{\Lambda^\prime}(r)] = (\zeta(m)V)^{-1} \text{\upshape vol } \mathcal{B}_r,$$ where the average is with respect to the uniform distribution on $\mathbb{L}_{p}$. \[thm:2\]

A slightly stronger version of the above result is precisely stated in Theorem \[thm:AverageEnsembleGeneral\]. We shall refer to a family of reductions that satisfies condition as *non-degenerate*. Non-degeneracy is indeed a very mild condition, and is satisfied, for instance, if $\Lambda_p$ has non-vanishing Hermite parameter (e.g., $\Lambda_p = p\mathbb{Z}^n$). Non-degenerate constructions immediately yield lattices satisfying the Minkowski-Hlawka lower bound (see the discussion in the end of Section \[sec:prelim\]). By choosing specific suitable families of non-degenerate reductions, 