An Algebraic Approach to Physical-Layer Network Coding

TLDR

The paper addresses designing new physical‑layer network coding (PNC) schemes using lattice partitions. It aims to demonstrate the potential of PNC in non‑asymptotic settings through an algebraic approach. The authors relate Nazer‑Gastpar’s PNC method to the fundamental theorem of finitely generated modules over a PID, generalize the code construction, simplify encoding and decoding, and apply the framework to a Gaussian relay network. The framework provides a transparent understanding of the original approach, enables the design of efficient practical PNC schemes, and outperforms conventional PNC in a Gaussian relay network.

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.