# Physical-layer Network Coding: A Random Coding Error Exponent Perspective

## TL;DR

This paper derives the error exponent for physical-layer network coding in two-way relay systems, providing insights and benchmarks for practical decoding performance with finite-length codes.

## Contribution

It introduces the first derivation of the random coding error exponent for PNC with sub-optimal XOR decoding, linking it to optimal decoding performance.

## Key findings

- Optimal PNC decoding error exponent can be achieved with sub-optimal XOR decoding.
- Provides bounds that serve as benchmarks for practical PNC system performance.
- Insights into finite-length code performance in PNC systems.

## Abstract

In this work, we derive the random coding error exponent for the uplink phase of a two-way relay system where physical layer network coding (PNC) is employed. The error exponent is derived for the practical (yet sub-optimum) XOR channel decoding setting. We show that the random coding error exponent under optimum (i.e., maximum likelihood) PNC channel decoding can be achieved even under the sub-optimal XOR channel decoding. The derived achievability bounds provide us with valuable insight and can be used as a benchmark for the performance of practical channel-coded PNC systems employing low complexity decoders when finite-length codewords are used.