Difficulties Constructing Lattices with Exponential Kissing Number from Codes
Metric Geometry
2025-07-28 v2 Information Theory
math.IT
Number Theory
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).
Cite
@article{arxiv.2410.16660,
title = {Difficulties Constructing Lattices with Exponential Kissing Number from Codes},
author = {Huck Bennett and Alexander Golovnev and Noah Stephens-Davidowitz},
journal= {arXiv preprint arXiv:2410.16660},
year = {2025}
}