  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
significantly smaller, in terms of alphabet-size, than previous ones in the
literature.
