Multistage Compute-and-Forward with Multilevel Lattice Codes Based on Product Constructions

TL;DR

This paper introduces a new lattice construction based on product codes, demonstrating their effectiveness for multistage compute-and-forward, and proposes practical constellations and decoding schemes with reduced complexity.

Contribution

It presents a novel product construction of lattices, proves their goodness properties, and develops a multistage compute-and-forward scheme with practical constellations and lower decoding complexity.

Findings

Lattices constructed via product codes are good for quantization and Poltyrev-good.

The proposed lattice codes achieve the same computation rate as existing schemes.

A new multistage decoding scheme reduces complexity based on the GCD of constellation size.

Abstract

A novel construction of lattices is proposed. This construction can be thought of as Construction A with codes that can be represented as the Cartesian product of $L$ linear codes over $\mathbb{F}_{p_1},\ldots,\mathbb{F}_{p_L}$, respectively; hence, is referred to as the product construction. The existence of a sequence of such lattices that are good for quantization and Poltyrev-good under multistage decoding is shown. This family of lattices is then used to generate a sequence of nested lattice codes which allows one to achieve the same computation rate of Nazer and Gastpar for compute-and-forward under multistage decoding, which is referred to as lattice-based multistage compute-and-forward. Motivated by the proposed lattice codes, two families of signal constellations are then proposed for the separation-based compute-and-forward framework proposed by Tunali \textit{et al.} together with a multilevel coding/multistage decoding scheme tailored specifically for these constellations. This scheme is termed separation-based multistage compute-and-forward and is shown having a complexity of the channel coding dominated by the greatest common divisor of the constellation size (may not be a prime number) instead of the constellation size itself.