# Raptor Codes for Higher-Order Modulation Using a Multi-Edge Framework

## TL;DR

This paper introduces a multi-edge framework for designing Raptor codes tailored for higher-order modulation, enabling improved performance through optimized degree distributions and joint belief propagation decoding.

## Contribution

It presents a novel multi-edge type LDPC code representation for Raptor codes and an efficient design method for higher-order modulation schemes.

## Key findings

- Designed Raptor codes outperform existing codes in higher-order modulation scenarios.
- Derived an exact stability condition for joint belief propagation decoding.
- Demonstrated the effectiveness of the multi-edge framework through multiple examples.

## Abstract

In this paper, we represent Raptor codes as multi-edge type low-density parity-check (MET-LDPC) codes, which gives a general framework to design them for higher-order modulation using MET density evolution. We then propose an efficient Raptor code design method for higher-order modulation, where we design distinct degree distributions for distinct bit levels. We consider a joint decoding scheme based on belief propagation for Raptor codes and also derive an exact expression for the stability condition. In several examples, we demonstrate that the higher-order modulated Raptor codes designed using the multi-edge framework outperform previously reported higher-order modulation codes in literature.