An Algebraic Approach to Physical-Layer Network Coding

TL;DR

This paper introduces an algebraic framework for physical-layer network coding that generalizes existing methods, simplifies encoding/decoding, and demonstrates practical advantages in Gaussian relay networks.

Contribution

It relates PNC to module theory, generalizes code construction, simplifies methods, and shows improved performance in practical network scenarios.

Findings

Algebraic framework relates PNC to module theory.

Simplified encoding and decoding methods.

Demonstrated advantage in Gaussian relay networks.

Abstract

The problem of designing new physical-layer network coding (PNC) schemes via lattice partitions is considered. Building on a recent work by Nazer and Gastpar, who demonstrated its asymptotic gain using information-theoretic tools, we take an algebraic approach to show its potential in non-asymptotic settings. We first relate Nazer-Gastpar's approach to the fundamental theorem of finitely generated modules over a principle ideal domain. Based on this connection, we generalize their code construction and simplify their encoding and decoding methods. This not only provides a transparent understanding of their approach, but more importantly, it opens up the opportunity to design efficient and practical PNC schemes. Finally, we apply our framework for PNC to a Gaussian relay network and demonstrate its advantage over conventional PNC schemes.