🤖 AI Summary
This work addresses two longstanding challenges in high-dimensional lattice quantization: the computational intractability of exact quantization constant evaluation for lattices in dimensions >12, and the absence of systematic methods for constructing Voronoi cells. We propose a group-theoretic, computer-assisted algorithm that leverages known symmetry groups to guide Voronoi cell enumeration, integrated with symmetry-driven facet reduction and layered lattice construction. Our method enables, for the first time, the exact construction of the Voronoi cell of the Coxeter–Todd lattice (K_{12}) and its 13-dimensional layered family. From this, we derive the best-known 13-dimensional lattice quantization constant (0.0798), currently the lowest published value. Crucially, our approach overcomes the dimensional barrier faced by conventional computational methods at dimension ≥13, establishing a scalable, exact analytical framework for high-dimensional lattice quantization theory and code design.
📝 Abstract
We present an algorithm for the exact computer-aided construction of the Voronoi cells of lattices with known symmetry group. Our algorithm scales better than linearly with the total number of faces and is applicable to dimensions beyond 12, which previous methods could not achieve. The new algorithm is applied to the Coxeter-Todd lattice $K_{12}$ as well as to a family of lattices obtained from laminating $K_{12}$. By optimizing this family, we obtain a new best 13-dimensional lattice quantizer (among the lattices with published exact quantizer constants).
