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.

To appear in IEEE Transactions on Communications. 10 pages