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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., Comment: 29 pages, 9 Figures, Submitted for possible publication to MDPI Journal Algorithms

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., Comment: This work has been submitted to the IEEE for possible publication

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 the Raptor code as a multi-edge type low-density parity-check (METLDPC) code. 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 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 correspond to inner code and 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. Finally we consider performance-complexity trade-offs of Raptor codes using the multi-edge framework. 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 the realized rate., Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible

This paper considers density evolution for lowdensity parity-check (LDPC) and multi-edge type low-density parity-check (MET-LDPC) codes over the binary input additive white Gaussian noise channel. We first analyze three singleparameter 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 to full density evolution thereby making it more suitable for code design.

A low-density parity-check (LDPC) code is a linear block code described by a sparse parity-check matrix, which can be efficiently represented by a bipartite Tanner graph. The standard iterative decoding algorithm, known as belief propagation, passes messages along the edges of this Tanner graph. Density evolution is an efficient method to analyze the performance of the belief propagation decoding algorithm for a particular LDPC code ensemble, enabling the determination of a decoding threshold. The basic problem addressed in this work is how to optimize the Tanner graph so that the decoding threshold is as large as possible. We introduce a new code optimization technique which involves the search space range which can be thought of as minimizing randomness in differential evolution or limiting the search range in exhaustive search. This technique is applied to the design of good irregular LDPC codes and multiedge type LDPC codes., Comment: 5 pages, 5 figures, conference paper

This paper considers the optimization of multi-edge type low-density parity-check (METLDPC) codes to maximize the decoding threshold. We propose an algorithm to jointly optimize 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 optimization is particularly important for MET-LDPC codes as it is not clear a priori which structures will be good. Using several examples, we demonstrate that the MET-LDPC codes designed by the proposed joint optimization algorithm exhibit improved decoding thresholds compared to previously reported MET-LDPC codes.