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

On the density of cyclotomic lattices constructed from codes