A new lower bound for Hermite's constant for symplectic lattices

In section 1 we give an improved lower bound on Hermite’s constant ${\delta }_{2g}$ for symplectic lattices in even dimensions ($g=2n$) by applying a mean-value argument from the geometry of numbers to a subset of symmetric lattices. Here we obtain only a slight improvement. However, we believe that the method applied has further potential. In section 2 we present new families of highly symmetric (symplectic) lattices, which occur in dimensions of powers of two. Here the lattices in dimension $2^n$ are constructed with the help of a multiplicative matrix group isomorphic to $({{\mathbb{Z}}_2}^n,+)$. We furthermore show the connection of these lattices with the circulant matrices and the Barnes-Wall lattices.