we can further improve this density and obtain a number of interesting corollaries. We highlight one of them:

\[cor:betterCyclotomic\] Let $\mathcal{O}_K$ be the ring of integers of a degree $n$ number field $K$ containing $r(K)$ roots of unity. For any integer $t\geq 2$, there exists an $\mathcal{O}_K$-lattice with dimension $t$ and packing density $$\Delta \geq \frac{r(K) t \zeta(tn)(1-\varepsilon)}{e(1-e^{-t}) 2^{tn}},$$ for any $\varepsilon > 0$.

This proves for instance, the existence of ideal lattices in any dimension with density better, by a linear factor, than the Minkowski-Hlawka lower bound. This also recovers, for $t=2$ and a judicious choice of number field and degree, the density in [@Venkatesh Thm. 1] and [@Moustrou Thm. 2]. By allowing reductions to codes over matrix rings (rather than the field $\mathbb{F}_p$), we provide, in Section \[sec:redRings\], a coding-theoretic proof of the existence of dense Hurwitz lattices, as in [@Vance].

Here is how Theorem 1 may be interpreted: the density of the kernel (coarse) lattice $\Lambda_p$ is improved by “adjoining” a code $\mathcal{C}$ to it, through the reduction $\phi_p$. Now if $\Lambda_p$ itself has a reasonable density, we can improve it up to the Minkowski-Hlawka bound. Building on this idea, we show in Section \[sec:conc\] that if 