  Recently, Venkatesh improved the best known lower bound for lattice sphere
packings by a factor $\log\log n$ for infinitely many dimensions $n$. Here we
prove an effective version of this result, in the sense that we exhibit, for
the same set of dimensions, finite families of lattices containing a lattice
reaching this bound. Our construction uses codes over cyclotomic fields, lifted
to lattices via Construction A.
