Constructing lattices with exponential kissing number

Construct an infinite family of lattices L_n ⊂ R^n whose Euclidean kissing number K_2(L_n) grows exponentially with the dimension n; specifically, achieve K_2(L_n) ≥ 2^{Ω(n)} for all n.

Background

The paper demonstrates that several natural constructions of lattices from error-correcting codes—such as Construction D and a "simplified construction D" using all embeddings of minimum-weight codewords—do not, in general, preserve a correspondence between minimum-weight codewords and shortest lattice vectors. As a result, prior claims by Vlăduţ that such constructions yield lattices with exponential kissing number are invalid, and those works have been retracted.

Establishing an explicit infinite family of lattices with exponentially many shortest vectors remains crucial both for geometric interest and for complexity-theoretic applications: certain hardness results for computational lattice problems rely on the existence of such families. The best known constructions to date have subexponential kissing numbers, leaving the exponential regime unresolved.