\mathcal{M}_2(\mathbb{F}_p)^m$ be a linear code with cardinality $|\mathcal{C}|$. Then $\Lambda_p^{\mathbb{H}}(\mathcal{C})=(\phi_p^{\mathbb{H}})^{-1}(\mathcal{C})$ is a quaternionic lattice with volume $|C| p^{-4m}$. Let $\mathcal{C}_b$ be a balanced set and $\mathbb{L}_p$ the associated lattice ensembles $$\mathbb{L}_p = \left\{ \beta \Lambda_p^{\mathbb{H}}(\mathcal{C}) :  \mathcal{C} \in \mathcal{C}_b \right\},$$ where $\beta = (V/(|C|^{-1} p^{4m}))^{1/(4m)}$. The following Theorem \[thm:matrices\] is the analogous of Theorem \[thm:2\] for Lipschitz lattices. We need the following lemma

If $\phi_p(x+yj)$ is non-invertible for $x+yi\in\mathcal{L}$, then the squared norm of $x+yj$ is a multiple of $p$. \[lem:1\]

If $\det \phi_p(x+yj) = 0$, then $\pi(x \overline{x} + y \overline{y}) = 0$, i.e., $\left\| (x,y)\right\|^2 \in \mathfrak{p}$. Since the norm of a Lipschitz quaternion is an integer, and $\mathfrak{p}$ is above $p$, the result follows.

Let $\mathcal{C}_b$ be a balanced set of codes with rank $k > m/2$. If $f$ is a semi-admissible function then $$\lim_{p\to\infty} E_{\mathbb{L}_p}\left[\sum_{{\mathbf{x}}\in \beta \Lambda_p^{\mathbb{H}}(\mathcal{C})}f(\psi({\mathbf{x}}))\right] \leq (\zeta(4m)V^{-1}) \int_{\mathbb{R}^{4m}} f({\mathbf{x}}) d{\mathbf{x}}\label{eq:LipschitzIntegers}$$ \[thm:matrices\]

The proof is very similar to that of Theorem \[thm:2\]. Here we divide the expectation into invertible and non-invertible elements (we make the change of variable ${\mathbf{x}}= \beta {\mathbf{y}}$, to facilitate), i.e. $$\sum_{{\mathbf{y}}\in \Lambda_p^{\mathbb{H}}(\mathcal{C})}f(\psi(\beta {\mathbf{y}})) = \sum_{{{\mathbf{y}}\in \Lambda_p^{\mathbb{H}}(\mathcal{C})} \above 0pt {\phi({{\mathbf{y}}}) \in (\mathcal{M}_2(\mathbb{F}_p)^m)^{*}}}f(\psi(\beta{\mathbf{y}}))+\sum_{{{\mathbf{x}}\in \Lambda_p^{\mathbb{H}}(\mathcal{C})} \above 0pt {\phi({{\mathbf{y}}}) \notin (\mathcal{M}_2(\mathbb{F}_p)^m)^{*}}}f(\psi(\beta{\mathbf{y}})).$$ The first term tends 