Number field lattices achieve Gaussian and Rayleigh channel capacity within a constant gap

TL;DR

This paper demonstrates that number field lattice codes can achieve near-capacity performance in both Gaussian and Rayleigh fading channels, with the gap determined by algebraic number theory properties.

Contribution

It introduces a family of lattice codes based on number fields that achieve a constant gap to capacity in diverse channel models, leveraging towers of Hilbert class fields.

Findings

Number field lattice codes achieve a constant gap to capacity in Gaussian and Rayleigh channels.

The gap is linked to the root discriminant of the number fields used.

Normalized minimum product distance in Rayleigh channels is analogous to Hermite invariant in Gaussian channels.

Abstract

This paper proves that a family of number field lattice codes simultaneously achieves a constant gap to capacity in Rayleigh fast fading and Gaussian channels. The key property in the proof is the existence of infinite towers of Hilbert class fields with bounded root discriminant. The gap to capacity of the proposed families is determined by the root discriminant. The comparison between the Gaussian and fading case reveals that in Rayleigh fading channels the normalized minimum product distance plays an analogous role to the Hermite invariant in Gaussian channels.