AWGN-Goodness is Enough: Capacity-Achieving Lattice Codes based on Dithered Probabilistic Shaping

TL;DR

This paper demonstrates that any lattice sequence good for the Gaussian channel can be shaped into capacity-achieving codes without additional lattice conditions, using discrete Gaussian distributions and dithering, especially effective at high SNR.

Contribution

It establishes a direct link between AWGN-goodness and capacity-achieving lattice codes without extra lattice constraints, utilizing discrete Gaussian properties and dithering.

Findings

Capacity-achieving lattice codes can be constructed from AWGN-good lattices.

The scheme approaches optimal dispersion at high SNR.

Dithering is necessary for low SNR performance.

Abstract

In this paper we show that any sequence of infinite lattice constellations which is good for the unconstrained Gaussian channel can be shaped into a capacity-achieving sequence of codes for the power-constrained Gaussian channel under lattice decoding and non-uniform signalling. Unlike previous results in the literature, our scheme holds with no extra condition on the lattices (e.g. quantization-goodness or vanishing flatness factor), thus establishing a direct implication between AWGN-goodness, in the sense of Poltyrev, and capacity-achieving codes. Our analysis uses properties of the discrete Gaussian distribution in order to obtain precise bounds on the probability of error and achievable rates. In particular, we obtain a simple characterization of the finite-blocklength behavior of the scheme, showing that it approaches the optimal dispersion coefficient for \textit{high} signal-to-noise ratio. We further show that for \textit{low} signal-to-noise ratio the discrete Gaussian over centered lattice constellations can not achieve capacity, and thus a shift (or "dither") is essentially necessary.