Glued lattices are better quantizers than $K_{12}$

TL;DR

The paper addresses the long‑standing question of the optimal 12‑dimensional lattice quantizer by challenging the status of the Coxeter–Todd lattice $K_{12}$. It applies gluing theory to unions of translated base lattices, notably $E_6\times E_6$ and $D_6\times D_6$, to produce new lattices whose normalized second moments (NSMs) are computed exactly. The key findings are two new 12‑D lattices with NSMs $G\approx 0.070058650$ (from $E_6\times E_6$) and $G\approx 0.070034$ (from $D_6\times D_6$), both beating $G_{K_{12}}\approx 0.070096$, with detailed structural properties and high symmetry. A byproduct yields an improved 13‑D quantizer with $G\approx 0.070974$, underscoring gluing as a powerful method for discovering superior lattice quantizers and suggesting further potential in other dimensions.

Abstract

40 years ago, Conway and Sloane proposed using the highly symmetrical Coxeter-Todd lattice $K_{12}$ for quantization, and estimated its second moment. Since then, all published lists identify $K_{12}$ as the best 12-dimensional lattice quantizer. Surprisingly, $K_{12}$ is not optimal: we construct two new 12-dimensional lattices with lower normalized second moments. The new lattices are obtained by gluing together 6-dimensional lattices.