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

2025-01-14

AI Summary

This work investigates the attainable ranges of the Rankin constant γ_{n,l} and the Bergé–Martinet constant γ'_{n,l} for lattices constructed from linear codes in the low-dimensional unsolved regime, specifically determining tight bounds for n = 5, 7 and l = 2. Method: Leveraging lattice theory, linear code constructions—particularly self-dual and extremal codes—combined with Minkowski’s theorem and volume-ratio analysis, we systematically derive rigorous bounds. Contribution/Results: We establish the first exact intervals: γ_{5,2} ∈ [√2, √3], γ'_{5,2} ∈ [1, √2], and strict bounds for both γ_{7,2} and γ'_{7,2}. These results improve several existing estimates for Rankin-type constants and, more significantly, forge a concrete connection between coding-theoretic lattice constructions and classical geometric lattice constants. Thereby, they extend the applicability of lattice constant theory to the realm of explicitly constructible lattices.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography

Abstract

The Rankin constant $gamma_{n,l}$ measures the largest volume of the densest sublattice of rank $l$ of a lattice $Lambdain RR^n$ 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 $gamma'_{5,2},gamma'_{7,2}$.

Problem

• Rankin Constants
• Bergé-Martinet Constants
• Lattice Cryptography

Innovation

• Rankin and Bergé-Martinet Constants
• Coding Theory
• Linear Codes Lattices