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

TL;DR

This paper investigates the Rankin and Bergé-Martinet constants for lattices derived from linear codes, providing bounds and revisiting known cases for dimensions 3, 4, 5, and 8, with new bounds for some open cases.

Contribution

It offers new bounds and insights on the Rankin and Bergé-Martinet constants specifically for code-based lattices, addressing open cases in dimensions 5 and 7.

Findings

Bounds for $oldsymbol{eta_{5,2}}$ and $oldsymbol{eta'_{5,2}}$ are established.

Bounds for $oldsymbol{eta_{7,2}}$ and $oldsymbol{eta'_{7,2}}$ are provided.

Revisits and refines known results for dimensions 3, 4, 8.

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 $\gamma'_{5,2},\gamma'_{7,2}$.