In this paper we introduce generalized Craig lattices, which allows us to construct lattices in Euclidean spaces of many dimensions in the range $3332-4096$ which are denser than the densest known Mordell-Weil lattices. Moreover we prove that if there were some nice linear binary codes we could construct lattices even denser in the range $128-3272$. We also construct some dense lattices of dimensions in the range $4098-8232$. Finally we also obtain some new lattices of moderate dimensions such as $68,84,85,86$, which are denser than the...

We consider the problem of constructing dense lattices in ${ℝ}^n$ with a given non trivial automorphisms group. We exhibit a family of such lattices of density at least $cn{2}^{-n}$, which matches, up to a multiplicative constant, the best known density of a lattice packing. For an infinite sequence of dimensions $n$, we exhibit a finite set of lattices that come with an automorphisms group of size $n$, and a constant proportion of which achieves the aforementioned lower bound on the largest packing density. The algorithmic...