Difficulties Constructing Lattices with Exponential Kissing Number from Codes
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 Vladut (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 Vladut 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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