Construction $\pi_A$ and $\pi_D$ Lattices: Construction, Goodness, and Decoding Algorithms

TL;DR

This paper introduces new lattice constructions, Construction π_A and π_D, demonstrating their effectiveness for channel coding and decoding in Gaussian channels, and generalizing existing lattice code frameworks.

Contribution

It proposes novel lattice constructions, analyzes their channel coding performance, and introduces a generalized construction encompassing previous methods.

Findings

Construction π_A lattices are good for channel coding under multistage decoding.

A new family of multilevel nested lattice codes based on Construction π_A is proposed.

Construction π_D generalizes several existing lattice constructions.

Abstract

A novel construction of lattices is proposed. This construction can be thought of as a special class of Construction A from codes over finite rings that can be represented as the Cartesian product of $L$ linear codes over $\mathbb{F}_{p_1},\ldots,\mathbb{F}_{p_L}$, respectively, and hence is referred to as Construction $\pi_A$. The existence of a sequence of such lattices that is good for channel coding (i.e., Poltyrev-limit achieving) under multistage decoding is shown. A new family of multilevel nested lattice codes based on Construction $\pi_A$ lattices is proposed and its achievable rate for the additive white Gaussian channel is analyzed. A generalization named Construction $\pi_D$ is also investigated which subsumes Construction A with codes over prime fields, Construction D, and Construction $\pi_A$ as special cases.