This paper considers density evolution for low-density parity-check (LDPC) and multi-edge type LDPC (MET-LDPC) codes over the binary input additive white Gaussian noise channel. We first analyze three single-parameter Gaussian approximations for density evolution and discuss their accuracy under several conditions, namely, at low rates, with punctured and degree-one variable nodes. We observe that the assumption of symmetric Gaussian distribution for the density-evolution messages is not accurate in the early decoding iterations, particularly at low rates and with punctured variable nodes. Thus, single-parameter Gaussian approximation methods produce very poor results in these cases. Based on these observations, we then introduce a new density evolution approximation algorithm for LDPC and MET-LDPC codes. Our method is a combination of full density evolution and a single-parameter Gaussian approximation, where we assume a symmetric Gaussian distribution only after density-evolution messages closely follow a symmetric Gaussian distribution. Our method significantly improves the accuracy of the code threshold estimation. Additionally, the proposed method significantly reduces the computational time of evaluating the code threshold compared with full density evolution thereby making it more suitable for code design.

The focus of this paper is on the analysis and design of Raptor codes using a multi-edge framework. In this regard, we first represent Raptor codes as multi-edge type (MET) low-density parity-check codes. This MET representation gives a general framework to analyze and design Raptor codes over a binary input additive white Gaussian noise channel using MET density evolution (MET-DE). We then consider a joint decoding scheme based on the belief propagation (BP) decoding for Raptor codes in the multi-edge framework, and analyze the convergence behavior of the BP decoder using MET-DE. In joint decoding of Raptor codes, the component codes corresponding to the inner code and the precode are decoded in parallel and provide information to each other. We also derive an exact expression for the stability of Raptor codes with joint decoding. We then propose an efficient Raptor code design method using the multi-edge framework, where we simultaneously optimize the inner code and the precode. Through density evolution analysis we show that the designed Raptor codes using the multi-edge framework outperform the existing Raptor codes in literature in terms of realized rates.

In this letter, we consider the design of Raptor codes for a wide signal-to-noise ratio (SNR) range using a multi-edge framework. The main idea is to change the degree distribution in a way to optimize the rate efficiency when more output bits are sent out. The proposed code shows near capacity performance in the designed SNR range, and achieves higher efficiencies compared to reference Raptor codes from the literature.

In this paper, we represent Raptor codes using a multi-edge framework, which gives a general framework to design them for higher-order modulation using multi-edge type 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.

This study considers the optimisation of multi‐edge type low‐density parity‐check (MET‐LDPC) codes to maximise the decoding threshold. The authors propose an algorithm to jointly optimise the node degree distribution and the multi‐edge structure of MET‐LDPC codes for given values of the maximum number of edge‐types and maximum node degrees. This joint optimisation is particularly important for MET‐LDPC codes as it is not clear a priori which structures will be good. Using several examples, they demonstrate that the MET‐LDPC codes designed by the proposed joint optimisation algorithm exhibit improved decoding thresholds compared with previously reported MET‐LDPC codes.

This work addresses the physical layer channel code design for an uncoordinated, frame- and slot-asynchronous random access protocol. Starting from the observation that collisions between two users yield very specific interference patterns, we define a surrogate channel model and propose different protograph low-density parity-check code designs. The proposed codes are both tested in a setup where the physical layer is abstracted, as well as on a more realistic channel model, where finite-length physical layer simulations of the entire asynchronous random access scheme, including decoding, are carried out. We find that the abstracted physical layer model overestimates the performance when short blocks are considered. Additionally, the optimized codes show gains in supported channel traffic, a measure of the number of terminals that can be concurrently accommodated on the channel, of around 17% at a packet loss rate of ${10}^{-2}$ w.r.t. off-the-shelf codes.