The alphabet ${𝐅}_3+v{𝐅}_3$ where $v^2=1$ is viewed here as a quotient of the ring of integers of $𝐐\left(\sqrt{-2}\right)$ by the ideal (3). Self-dual ${𝐅}_3+v{𝐅}_3$ codes for the hermitian scalar product give $2$-modular lattices by construction $A_K$. There is a Gray map which maps self-dual codes for the Euclidean scalar product into Type III codes with a fixed point free involution in their automorphism group. Gleason type theorems for the symmetrized weight enumerators of Euclidean self-dual codes and the length weight enumerator of hermitian self-dual...

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...