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

TL;DR

This paper demonstrates that newly constructed glued lattices outperform the traditional Coxeter-Todd lattice $K_{12}$ as quantizers in 12 dimensions by achieving lower normalized second moments.

Contribution

It introduces a novel method of constructing 12-dimensional lattices through gluing, resulting in better quantization performance than the long-standing $K_{12}$ lattice.

Findings

New lattices have lower normalized second moments than $K_{12}$

Gluing 6-dimensional lattices is effective for improving quantization

Challenged the long-held belief of $K_{12}$ optimality

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.