About the Rankin and Berg\'e-Martinet Constants from a Coding Theory View Point

Abstract

The Rankin constant $\gamma_{n,l}$ measures the largest volume of the densest sublattice of rank $l$ of a lattice $\Lambda\in \RR^n$ over all such lattices of rank $n$. The Berg\'e-Martinet constant $\gamma'_{n,l}$ is a variation that takes into account the dual lattice. Exact values and bounds for both constants are mostly open in general. We consider the case of lattices built from linear codes, and look at bounds on $\gamma_{n,l}$ and $\gamma'_{n,l}$. In particular, we revisit known results for $n=3,4,5,8$ and give lower and upper bounds for the open cases $\gamma_{5,2},\gamma_{7,2}$ and $\gamma'_{5,2},\gamma'_{7,2}$.