# Asymptotic Analysis on LDPC-BICM Scheme for Compute-and-Forward Relaying

## TL;DR

This paper analyzes the asymptotic decoding performance of LDPC-BICM schemes in compute-and-forward relay channels, demonstrating higher achievable rates for QPSK and 8PSK modulations compared to separation decoding.

## Contribution

It introduces an asymptotic analysis of LDPC-BICM schemes for CAF using density evolution without the all-zero codeword assumption.

## Key findings

- LDPC-BICM achieves higher rates than separation decoding.
- Density evolution with population dynamics effectively analyzes asymmetric channels.
- QPSK and 8PSK modulations benefit from the proposed scheme.

## Abstract

The compute-and-forward (CAF) scheme has attracted great interests due to its high band-width efficiency on two-way relay channels. In the CAF scheme, a relay attempts to decode a linear combination of transmitted messages from other terminals or relays. It is a crucial issue to study practical error-correcting codes in order to realize the CAF scheme with low computational complexity. In this paper, we present an efficient bit-interleaved coded modulation (BICM) scheme for the CAF scheme with phase shift keying (PSK) modulations. In particular, we examine the asymptotic decoding performance of the BICM scheme with low-density parity-check (LDPC) codes by using the density evolution (DE) method. Based on the asymmetric nature of the channel model, we utilize the population dynamics method for the DE equations without the all-zero codeword assumption. The results show that, for two-way relay channels with QPSK and 8PSK modulations, the LDPC-BICM scheme provides higher achievable rate compared with an alternative separation decoding scheme.