---
abstract: |
    We propose a general framework to study constructions of Euclidean lattices from linear codes over finite fields. In particular, we prove general conditions for an ensemble constructed using linear codes to contain dense lattices (i.e., with packing density comparable to the Minkowski-Hlawka lower bound). Specializing to number field lattices, we obtain a number of interesting corollaries - for instance, the best known packing density of ideal lattices, and an elementary coding-theoretic construction of asymptotically dense Hurwitz lattices. All results are algorithmically effective, in the sense that, for any dimension, a finite family containing dense lattices is exhibited. For suitable constructions based on Craig’s lattices, this family is smaller, in terms of alphabet-size, than previous ones in the literature.\
    **Keywords:** Lattices, sphere packings, random codes, ideal lattices, codes over matrix rings
author:
- 'Antonio Campello [^1]'
bibliography:
- 'campbel.bib'
title: |
    Random Ensembles of Lattices\
    from Generalized Reductions
---

Introduction
============

There has been a renewed interest in the search for new constructions of lattices from error-correcting codes due to their various recent applications, such as coding for fading wiretap channels [@KosiOngOggier], Gaussian relay networks [@Adaptative], compound fading channels [@Our] and index codes [@Index], to name only a 