A Simpler Proof on the Existence of Good Nested Lattice Codes Over Imaginary Quadratic Integers for AWGN Channel

Abstract

This paper investigates nested lattice codes generated through Construction A from the ring of integers of an imaginary quadratic field. Our primary goal is to offer a streamlined proof of the existence of nested lattice codes that can attain the capacity of an Additive White Gaussian Noise (AWGN) channel. We alter the random ensemble of nested lattice codes by introducing discrete random dithers instead of continuous random dithers. This adjustment enables us to draw a parallel between nested lattice codes and nested linear codes, facilitating a proof that remains as straightforward as that used for nested linear codes. Furthermore, we demonstrate that this collection of lattices exhibits favorable properties for Mean Square Error (MSE) quantization under specific constraints.

Keywords

• Algebraic integers
• AWGN channel
• lattice codes
• MSE quantization