We design low-density parity-check (LDPC) codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on [1]. Assuming that the underlying communication channel is symmetric, we prove that the probability densities at the message nodes of the graph possess a certain symmetry. Using this symmetry property we then show that, under the assumption of no cycles, the message densities always converge as the number of iterations tends to infinity. Furthermore, we prove a stability condition which implies an upper bound on the fraction of errors that a belief-propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution. Our codes are found by optimizing the degree structure of the underlying graphs. We develop several strategies to perform this optimization. We also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds.

A numerical method has recently been presented to determine the noise thresholds of low density parity-check (LDPC) codes that employ the message passing decoding algorithm on the additive white Gaussian noise (AWGN) channel. In this paper, we extend this technique to the uncorrelated flat Rayleigh fading channel. Using a nonlinear code optimization technique, we optimized irregular LDPC codes for the uncorrelated Rayleigh fading channel. The thresholds of the optimized irregular LDPC codes are very close to the Shannon limit for this channel. For example, at rate one-half, the optimized irregular LDPC code has a threshold only 0.07dB away from the capacity of this channel. Furthermore, we compare simulated performance of the optimized irregular LDPC codes and turbo codes on a land mobile channel, and the results indicate that at a block size of 3072, irregular LDPC codes can outperform turbo codes over a wide range of mobile speeds. This work was sponsored by the National Science Fo...

We design sequences of low-density parity check codes that provably perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on [1]. Additionally, based on the assumption that the underlying communication channel is symmetric, we prove that the probability densities at the message nodes of the graph satisfy a certain symmetry. This enables us to derive a succinct description of the density evolution for the case of a belief propagation decoder. Furthermore, we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution. Our codes are found by optimizing the degree structure of the underlying graphs. We develop several strategies to perform this optimization. We also present s...

Submitted to the Proceedings of the IEEE

We present an optimized channel coding scheme for OFDM transmitter. Traditional coding methods use regular codes, in the sense that each bit participates in the same way to the channel encoding. Our approach consists in using a priory assumption on the channel available at the transmitter in order to optimize the coding scheme. We have considered the irregular Gallager block codes in our study. Simulations provide evidence of the usefulness of our approach with a gain of 2 dB at a bit error rate equal to ## for optimized irregular coding scheme compared to regular one.

Abstract—We apply low-density parity-check (LDPC) codes to a bandwidth-efficient modulation scheme using multilevel coding, multistage decoding, and trellis-based signal shaping. Performance results based on density evolution and simulations are presented. Using irregular LDPC component codes of block length IHS and a 64-quadrature amplitude modulation signal constellation operating at 2 bits/dimension, a bit-error rate of IH S is achieved at an H of 6.55 dB. At this value of H, the Shannon channel capacity, computed assuming equally likely signaling, is below 2 bits/dimension. Index Terms—Density evolution, low-density parity-check (LDPC) codes, multilevel coding (MLC), trellis shaping.

Abstract—Reconciliation is an essential part of any secret-key agreement protocol and hence of a Quantum Key Distribution (QKD) protocol, where two legitimate parties are given correlated data and want to agree on a common string in the presence of an adversary, while revealing a minimum amount of information. In this paper, we show that for discrete-variable QKD protocols, this problem can be advantageously solved with Low Density Parity Check (LDPC) codes optimized for the BSC. In particular, we demonstrate that our method leads to a significant improvement of the achievable secret key rate, with respect to earlier interactive reconciliation methods used in QKD. I.

To my parents, family, and Michael ii ACKNOWLEDGEMENTS I would like to thank the many people whose support has made the completion of my Ph.D. possible.

This paper starts a systematic study of capacityachieving sequences of low-density parity-check codes for the erasure channel. We introduce a class A of analytic functions and develop a procedure to obtain degree distributions for the codes. We show various properties of this class which will help us to construct new distributions from old ones. We then study certain types of capacity-achieving sequences and introduce new measures for their optimality. For instance, it turns out that the right-regular sequence is capacity-achieving in a much stronger sense than, e.g., the Tornado sequence. This also explains why numerical optimization techniques tend to favor graphs with only one degree of check nodes.

Pulse-position modulation (PPM) using a photon-counting receiver produces an extremely sensitive optical communications system, capable of transmitting multiple bits of information for each received photon. Such impressive sensitivity requires powerful errorcorrection codes that must be computationally efficient to enable high data throughput. Fountain codes combine performance and efficiency for a narrow class of channels, known as erasure channels. A potential application for fountain codes is the photon-counting PPM receiver, which is an approximate erasure channel and includes occasional channel errors. This nonideal behavior requires a nontraditional use of