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

TL;DR

This paper improves the lower bound of Hermite's constant for symplectic lattices in even dimensions using a mean-value approach and introduces new highly symmetric lattice families related to circulant matrices and Barnes-Wall lattices.

Contribution

It provides a slight improvement on Hermite's constant lower bound and constructs new symmetric lattice families in dimensions that are powers of two.

Findings

Slight improvement in Hermite's constant lower bound for symplectic lattices.

Construction of new symmetric lattices using multiplicative matrix groups.

Connection established between these lattices, circulant matrices, and Barnes-Wall lattices.

Abstract

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.