Difficulties Constructing Lattices with Exponential Kissing Number from Codes

TL;DR

This paper demonstrates that common methods of constructing lattices from error-correcting codes do not reliably produce lattices with exponential kissing number, invalidating previous key results and leaving the problem open.

Contribution

It provides counterexamples showing the failure of certain lattice constructions from codes to achieve exponential kissing number, challenging prior claims in the literature.

Findings

Counterexamples invalidate previous lattice construction results.

Constructing lattices with exponential kissing number remains unresolved.

Previous proofs claiming such constructions are flawed.

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).