few. For the applications considered in these works, it is desirable to lift a code over a finite field into a lattice that possesses a rich algebraic structure, often inherited from the properties of number fields. In the present work we provide an unified analysis of these constructions and investigate “random-coding”-like results for such lattices. Our focus is on the problem of finding dense structured lattice packings, although our techniques have a much broader scope of applications (as discussed in Section \[sec:conclusion\]).

Indeed, finding the densest packing is a central subject in the Geometry of Numbers, with a variety of well-established connections to Coding Theory. Let $\Delta_n$ denote the best possible sphere packing density achievable by a Euclidean lattice of dimension $n$. The celebrated Minkowski-Hlawka theorem (e.g. [@Cassels; @Leker]) gives the lower bound $\Delta_n \geq \zeta(n)/2^{n-1}$ for all $n\geq 2$, where $\zeta(n) = 1+1/2^n+1/3^n+\ldots$ is the Riemann zeta function. Up to very modest asymptotic improvements, this is still, to date, the best lower bound for the best packing density in high dimensions.

Typical methods for establishing the theorem depend on the construction of random ensembles of lattices and on mean-value results [@Siegel; @Rogers]. Rush [@Rush1989] and Loeliger [@Loeliger] obtained the 