About the Rankin and Bergé-Martinet Constants from a Coding Theory View Point
Frédérique Oggier, Shengwei Liu, Hongwei Liu
TL;DR
The article investigates the Rankin constants $\gamma_{n,l}$ and Bergé-Martinet constants $\gamma'_{n,l}$ for lattices, focusing on lattices arising from linear codes via Construction A. It develops general bounds that depend on code parameters and duality, and derives explicit formulas such as $\gamma'_{n,1}(\Lambda_C)=\tfrac{1}{q}\sqrt{2\min(n,q^2)}$ and $\gamma'_{n,2}(\Lambda_C)=\tfrac{1}{q^2}\sqrt{d_2(\Lambda_C)d_2(\Lambda_C^*)}$, while connecting these invariants to the dual code via $\Lambda_C^*$ and $\Lambda_{C^{\perp}}$. The paper provides concrete results for dimensions $n=3,4,5,8$ and offers bounds for open cases $\gamma_{5,2},\gamma_{7,2},\gamma'_{5,2},\gamma'_{7,2}$, including realizations through $D_n$ lattices and Reed–Mueller constructions that relate to $E_8$. These findings advance the understanding of extremal lattice invariants from code-based constructions and bear implications for lattice-based cryptography and duality in coding theory.
Abstract
The Rankin constant $γ_{n,l}$ measures the largest volume of the densest sublattice of rank $l$ of a lattice $Λ\in \RR^n$ over all such lattices of rank $n$. The Bergé-Martinet constant $γ'_{n,l}$ 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 $γ_{n,l}$ and $γ'_{n,l}$. In particular, we revisit known results for $n=3,4,5,8$ and give lower and upper bounds for the open cases $γ_{5,2},γ_{7,2}$ and $γ'_{5,2},γ'_{7,2}$.