  Using graph-theoretic methods we give a new proof that for all sufficiently
large $n$, there exist sphere packings in $\R^n$ of density at least
$cn2^{-n}$, exceeding the classical Minkowski bound by a factor linear in $n$.
This matches up to a constant the best known lower bounds on the density of
sphere packings due to Rogers, Davenport-Rogers, and Ball. The suggested method
makes it possible to describe the points of such a packing with complexity
$\exp(n\log n)$, which is significantly lower than in the other approaches.
