lower bound from integer lattices constructed from linear codes in $\mathbb{F}_p^n$, in the limit when $p\to\infty$, with random-coding arguments. Improvements of the lower bound, in turn, strongly rely on additional (algebraic) structure. For instance, Vance [@Vance] showed that the best quaternionic lattice in dimension $m$ (with real equivalent in dimension $4m$) has density at least $3m \zeta(4m)/e 2^{4m-3}\leq \Delta_{4m}$. Using lattices built from cyclotomic number fields, Venkatesh [@Venkatesh] established the bound $\Delta_{2\varphi(m)} \geq m/2^{2\varphi(m)-1}$, where $\varphi(m)$ is Euler’s totient function (since $m$ can grow as fast as a constant times $\varphi(m)\log \log \varphi(m)$ this provides the first super-linear improvement). From another perspective, Gaborit and Zémor [@Zemor], and Moustrou [@Moustrou] exploited additional coding-theoretic and algebraic structures to significantly reduce the family size of ensembles containing dense lattices.\
**Main Contributions.** In this work we investigate general random lattices obtained from error correcting codes. The objective of this study is two-fold: we provide unified analyses and coding-theoretic proofs of the aforementioned results, as well as a simple condition to verify if any new construction can be used to build ensembles containing dense lattices. We start from the fairly general definition of a *reduction*, i.e. a mapping that takes a lattice into the space 