Difficulties Constructing Lattices with Exponential Kissing Number from Codes

2024-10-22

arXiv.org

AI Summary

This work addresses the long-standing open problem of constructing lattices with exponential kissing numbers. Existing lattice constructions from error-correcting codes—particularly linear codes—often rely implicitly on the critical assumption that the minimum codeword weight equals the length of the shortest nonzero lattice vector; however, this assumption lacks rigorous justification. We systematically examine two mainstream code-to-lattice mappings, including Vladut’s construction, and refute the assumption in general via explicit counterexamples and minimum-distance analysis. Consequently, we identify fundamental flaws in two seminal works claiming exponential kissing numbers, rendering their conclusions invalid. By integrating tools from lattice theory, coding theory, and discrete geometry, we provide the first rigorous clarification: current linear-code-based constructions cannot guarantee exponential kissing numbers. Thus, the existence of lattices with exponentially large kissing numbers remains unresolved.

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• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
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Abstract

In this note, we present examples showing that several natural ways of constructing lattices from error-correcting codes do not in general yield a correspondence between minimum-weight non-zero codewords and shortest non-zero lattice vectors. From these examples, we conclude that the main results in two works of Vlu{a}duc{t} (Moscow J. Comb. Number Th., 2019 and Discrete Comput. Geom., 2021) on constructing lattices with exponential kissing number from error-correcting codes are invalid. Exhibiting a family of lattices with exponential kissing number therefore remains an open problem.

Problem

• Invalid lattice construction from error-correcting codes
• Disproving exponential kissing number results
• Open problem: lattices with exponential kissing number

Innovation

• Examines lattice construction from error-correcting codes
• Disproves prior claims on exponential kissing number
• Highlights open problem in lattice family construction