# Compute-and-forward relaying with LDPC codes over QPSK scheme

## TL;DR

This paper analyzes the decoding performance of compute-and-forward relaying with LDPC codes over QPSK, showing it can outperform traditional schemes in high-rate scenarios with manageable complexity.

## Contribution

It provides a theoretical analysis of CAF with LDPC codes under QPSK, including performance comparison with separation decoding schemes and practical feasibility insights.

## Key findings

- CAF with LDPC codes achieves higher rates than traditional schemes.
- Decoding complexity of CAF is comparable to SIC and lower than joint decoding.
- CAF outperforms SD in high-rate regions based on density evolution results.

## Abstract

In this paper, we study a compute-and-forward (CAF) relaying scheme with low-density parity-check (LDPC) codes, a special case of physical layer network coding, under the quadrature phase shift keying (QPSK) modulation. The novelty of this paper is the theoretical analysis of decoding performance of the CAF scheme and traditional separation decoding (SD) scheme with joint decoding or with successive interference cancellation (SIC) decoding when the reception powers from both senders are not equal but close to each other. First, we study the decoding performance of linear random coding (LRC) in the CAF scheme whose decoder is based on the degraded channel. When rotation angles of constellations of two senders are controlled, we show that they can achieve rates well beyond the multiple access channel (MAC) with LRC with optimal rotation angles. Second, we analyze the practical feasibility of CAF using LDPC codes in terms of computational costs and decoding performance of belief propagation (BP) decoders. The calculation complexity of the BP decoder for the CAF scheme is almost equal to that of the SIC BP decoder, and smaller than the joint BP decoder in the SD scheme. Decoding performance is evaluated by the asymptotic decodable region using density evolution. The results show that, with code rate fixed, the CAF scheme has better performance than the SD scheme with joint BP decoding and SIC BP decoding in the high rate region.