Abstract
In this paper we give an improved lower bound on Hermite's constant δ2g 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. We obtain only a slight improvement. However, we believe that the method applied has further potential. Furthermore, in this paper we present new families of highly symmetric (symplectic) lattices, which occur in dimensions of powers of two. The lattices in dimension 2n are constructed with the help of a multiplicative matrix group isomorphic to (ℤ2n, +). We furthermore show the connection of these lattices with the circulant matrices and the Barnes-Wall lattices.