proving the statement.

Lipschitz and Hurwitz Lattices
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The quaternion skew-field $\mathbb{H}$ is given by $\mathbb{H} = \left\{ a+bi + (c+di)j: a, b, c, d \in \mathbb{R}\right\}$, with the usual relations $i^2=j^2=-1$ and $ij = -ji$. Vance recently [@Vance] proved a Minkowski-Hlawka theorem for lattices in $\mathbb{H}$ over the Hurwitz order. Here we show how to recover a “coding-theoretic” version of this result from generalized reductions.

We first explain how to deduce a slightly simpler case, for the Lipschitz order. The *Lipschitz integers* $\mathcal{L} \subset \mathbb{H}$ is the (non-maximal) order $\mathcal{L}=\left\{x+yj : x, y \in \mathbb{Z}[i]\right\}.$ Recall that a quaternion has matrix representation $$\left(\begin{array}{cc} x & -\overline{y} \\ y & \overline{x} \end{array}\right).$$ Let $\mathfrak{p}$ be an ideal in $\mathbb{Z}[i]$ above $p$ that splits. Let $\pi:\mathbb{Z}[i]\to\mathbb{Z}[i]/\mathfrak{p}$ be a projection. We consider the following “single-letter” reduction: $$\phi_p^{\mathbb{H}}:\mathcal{L} \to \mathcal{M}_2(\mathbb{F}_p)$$ $$\phi_p(x+yj)^{\mathbb{H}}=\left(\begin{array}{cc} \pi(x) & -\pi(\overline{y}) \\ \pi(y) & \pi(\overline{x}) \end{array}\right).$$ We have $\ker \phi_p^{\mathbb{H}} = (p\mathbb{Z}[i])+(p\mathbb{Z}[i])j$. Identifying $\mathbb{H}$ with $\mathbb{R}^4$ in the natural way $$\psi(a,b,c,d) \to a+bi + (c+di)j$$ we obtain a reduction $\phi_p:\mathbb{Z}^4 \to \mathcal{M}_2(\mathbb{F}_p)$, $\phi_p(x) = \phi_p^{\mathbb{H}}(\psi(x)).$ By abuse of notation, we will also denote by $\phi_p^{\mathbb{H}}$ the reduction applied componentwise in vector of $\mathcal{L}^m$, i.e., $$\phi_p^{\mathbb{H}}(x_1+y_1j,\ldots,x_m+y_mj) = (\phi_p^{\mathbb{H}}(x_1+y_1j),\ldots,\phi_p^{\mathbb{H}}(x_m+y_mj))\in\mathcal{M}_2(\mathbb{F}_p)^m.$$ Let $\mathcal{C} \subset 