Better Lattice Quantizers Constructed from Complex Integers

TL;DR

This paper introduces new low-dimensional complex lattice quantizers based on Eisenstein and Gaussian integers, achieving the best known quantizers in several dimensions and providing fast algorithms for their evaluation.

Contribution

It constructs novel complex lattice quantizers from Eisenstein and Gaussian integers and links their bases to cosets, reporting the best known quantizers in multiple dimensions.

Findings

Best known lattice quantizers in dimensions 14, 15, 18, 19, 22, 23

Proposes fast algorithms for quantization of generalized checkerboard lattices

Enables efficient evaluation of normalized second moment (NSM) via Monte Carlo

Abstract

This paper investigates low-dimensional quantizers from the perspective of complex lattices. We adopt Eisenstein integers and Gaussian integers to define checkerboard lattices $\mathcal{E}_{m}$ and $\mathcal{G}_{m}$. By explicitly linking their lattice bases to various forms of $\mathcal{E}_{m}$ and $\mathcal{G}_{m}$ cosets, we discover the $\mathcal{E}_{m,2}^+$ lattices, based on which we report the best known lattice quantizers in dimensions $14$, $15$, $18$, $19$, $22$ and $23$. Fast quantization algorithms of the generalized checkerboard lattices are proposed to enable evaluating the normalized second moment (NSM) through Monte Carlo integration.