Better Lattice Quantizers Constructed from Complex Integers

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.