  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 $({\Z_2}^n,+)$. We furthermore show
the connection of these lattices with the circulant matrices and the
Barnes-Wall lattices.
