Better Lattice Quantizers Constructed from Complex Integers

Abstract

Real-valued lattices and complex-valued lattices are mutually convertible, thus we can take advantages of algebraic integers to defined good lattice quantizers in the real-valued domain. In this paper, we adopt complex integers to define generalized checkerboard lattices, especially $\mathcal{E}_{m}$ and $\mathcal{E}_{m}^+$ defined by Eisenstein integers. Using $\mathcal{E}_{m}^+$, we report the best lattice quantizers in dimensions $14$, $18$, $20$, and $22$. Their product lattices with integers $\mathbb{Z}$ also yield better quantizers in dimensions $15$, $19$, $21$, and $23$. The Conway-Sloane type fast decoding algorithms for $\mathcal{E}_{m}$ and $\mathcal{E}_{m}^+$ are given.Comment: 7 page