The gap between shortest and second-shortest vectors in Barnes-Wall lattices is at least 50%.
Nebe (arXiv:2603.23133) proves that the second minimum of the Barnes-Wall lattices is at least 3/2 of the minimum. The Barnes-Wall lattices form an infinite family — one in each dimension 2^n — that appear in coding theory, signal processing, and sphere packing. They include celebrated examples: the hexagonal lattice in dimension 2, the Gosset lattice E_8 in dimension 8, the Leech lattice in dimension 24 (up to scaling).
The minimum distance of a lattice — the length of its shortest nonzero vector — determines its packing density: how tightly spheres of that radius can be arranged. The second minimum determines the first shell beyond the kissing number. The gap between them constrains the lattice's error-correction properties, its robustness as a quantization scheme, and the spectral gap of associated theta functions.
A ratio of 3/2 means that any lattice vector shorter than 3/2 of the minimum must itself be a minimum-length vector. The lattice has no vectors at intermediate distances. This clean gap — no vectors in the interval between the minimum and 3/2 of the minimum — is a structural rigidity property: the lattice doesn't allow near-minimum vectors that might blur the distinction between closest and second-closest.
For a family spanning all power-of-two dimensions, a uniform bound on this ratio is rare. Most lattice families have second minima that depend on dimension in complicated ways. The Barnes-Wall family maintains its gap universally.