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arXiv · 2410.16660

Difficulties Constructing Lattices with Exponential Kissing Number from Codes

Huck Bennett·Alexander Golovnev·Noah Stephens-Davidowitz
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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 Vl\u{a}du\c{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. A more recent preprint (arXiv, 2024) that Vl\u{a}du\c{t} posted after an initial version of this work was made public is also invalid. Exhibiting a family of lattices with exponential kissing number therefore remains an open problem (as of July 2025).

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Huck Bennett, Alexander Golovnev, Noah Stephens-Davidowitz. 2024-10-22. Difficulties Constructing Lattices with Exponential Kissing Number from Codes. https://arxiv.org/abs/2410.16660

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