Abstract—We develop improved algorithms to construct good low-density parity-check codes that approach the Shannon limit very closely. For rate 1/2, the best code found has a threshold within 0.0045 dB of the Shannon limit of the binary-input additive white Gaussian noise channel. Simulation results with a somewhat simpler code show that we can achieve within 0.04 dB of the Shannon limit at a bit error rate of 10 T using a block length of 10 U. Index Terms—Density evolution, low-density parity-check codes, Shannon limit, sum-product algorithm. I.

Density evolution is an algorithm for computing the capacity of low-density parity-check (LDPC) codes under messagepassing decoding. For memoryless binary-input continuous-output additive white Gaussian noise (AWGN) channels and sum-product decoders, we use a Gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm. We convert the infinite-dimensional problem of iteratively calculating message densities, which is needed to find the exact threshold, to a one-dimensional problem of updating means of Gaussian densities. This simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better, but also makes it easier to design good irregular LDPC codes for AWGN channels. For various regular LDPC codes we have examined, thresholds can be estimated within 0.1 dB of the exact value. For rates between 0.5 and 0.9, codes designed using the Gaussian approximation perform within 0.02 dB of the best performing codes found so far by using density evolution when the maximum variable degree is IH. We show that by using the Gaussian approximation, we can visualize the sum-product decoding algorithm. We also show that the optimization of degree distributions can be understood and done graphically using the visualization.

Abstract—Extrinsic information transfer (EXIT) charts are a tool for predicting the convergence behavior of iterative processors for a variety of communication problems. A model is introduced that applies to decoding problems, including the iterative decoding of parallel concatenated (turbo) codes, serially concatenated codes, low-density parity-check (LDPC) codes, and repeat–accumulate (RA) codes. EXIT functions are defined using the model, and several properties of such functions are proved for erasure channels. One property expresses the area under an EXIT function in terms of a conditional entropy. A useful consequence of this result is that the design of capacity-approaching codes reduces to a curve-fitting problem for all the aforementioned codes. A second property relates the EXIT function of a code to its Helleseth–Kløve–Levenshtein information functions, and thereby to the support weights of its subcodes. The relation is via a refinement of information functions called split information functions, and via a refinement of support weights called split support weights. Split information functions are used to prove a third property that relates the EXIT function of a linear code to the EXIT function of its dual. Index Terms—Concatenated codes, duality, error-correction coding, iterative decoding, mutual information.

In this correspondence, we consider puncturing of low-density parity-check (LDPC) codes for additive white Gaussian noise (AWGN) channels. We show that good puncturing patterns exist and that the puncturing can be performed in a rate-compatible fashion. Furthermore, ratecompatible puncturing results in a small loss of performance with respect to threshold, namely, the punctured code is good (in terms of threshold) across a range of rates when compared with the optimal codes for each rate. This allows one to implement a single “mother” encoder and decoder that is good across a wide range of rates.

We optimize the random-like ensemble of Irregular Repeat Accumulate (IRA) codes for binary-input symmetric channels in the large blocklength limit. Our optimization technique is based on approximating the Evolution of the Densities (DE) of the messages exchanged by the Belief-Propagation (BP) message-passing decoder by a one-dimensional dynamical system. In this way, the code ensemble optimization can be solved by linear programming. We propose four such DE approximation methods, and compare the performance of the obtained code ensembles over the binary symmetric channel (BSC) and the binaryantipodal input additive white Gaussian channel (BIAWGNC). Our results clearly identify the best among the proposed methods and show that the IRA codes obtained by these methods are competitive with respect to the best-known irregular Low-Density Parity-Check codes (LDPC). In view of this and the very simple encoding structure of IRA codes, they emerge as attractive design choices.

Abstract — We use systematic irregular repeat accumulate (IRA) codes as source-channel codes for the transmission of an equiprobable memoryless binary source with side information at the decoder. A special case of this problem is joint source-channel coding for a nonequiprobable memoryless binary source. The theoretical limits of this problem are given by combining the Slepian-Wolf theorem, the source entropy in the special case, with the channel capacity. The approach is based on viewing the correlation between the binary source output and the side information as a separate channel or an enhancement of the original channel. The joint source-channel encoding, decoding and code design procedures are explained in detail. The simulated performance results are better than the recently published solutions using turbo codes and very close to the theoretical limit. I.

Abstract — A construction method for protograph-based LDPC codes that simultaneously achieve low iterative decoding threshold and linear minimum distance is proposed. We start with a high-rate protograph LDPC code with variable node degrees of at least 3. Lower rate codes are obtained by splitting check nodes and connecting them by degree-2 nodes. This guarantees the linear minimum distance property for the lower-rate codes. Excluding checks connected to degree-1 nodes, we show that the number of degree-2 nodes should be at most one less than the number of checks for the protograph LDPC code to have linear minimum distance. Iterative decoding thresholds are obtained by using the reciprocal channel approximation. Thresholds are lowered by using either precoding or at least one very highdegree node in the base protograph. A family of high- to low-rate codes with minimum distance linearly increasing in block size and with capacity-approaching performance thresholds is presented. FPGA simulation results for a few example codes show that the proposed codes perform as predicted. I.

The performance loss due to separation of detection and decoding on the binary-input additive white Gaussian noise channel is quantified in terms of mutual information. Results are reported for both the code-division multiple-access (CDMA) channel in the large system limit and the intersymbol interference (ISI) channel. The results for CDMA rely on the replica method developed in statistical mechanics. It is shown that a previous result in [1] found for Gaussian input alphabet holds also for binary input alphabets. For the ISI channel, the performance loss is calculated via the BCJR algorithm. Comparisons are made to the capacity of separate detection and decoding using suboptimum detectors such as a decision-feedback equalizer.

In this paper, we consider the problem of robust joint source-channel coding over an additive white Gaussian noise channel. We propose a new scheme which achieves the optimal slope of the signal-todistortion (SDR) curve (unlike the previously known coding schemes). Also, we propose a family of robust codes which together maintain a bounded gap with the optimum SDR curve (in terms of dB). To show the importance of this result, we drive some theoretical bounds on the asymptotic performance of delay-limited hybrid digital-analog (HDA) coding schemes. We show that, unlike the delay-unlimited case, for any family of delay-limited HDA codes, the asymptotic performance loss is unbounded (in terms of dB).

This paper focuses on finite-dimensional upper and lower bounds on decodable thresholds of Zm and binary LDPC codes, assuming belief propagation decoding on memoryless channels. Two noise measures will be considered: the Bhattacharyya noise parameter and the soft bit value for a MAP decoder on the uncoded channel. For Zm LDPC codes, an iterative m-dimensional bound is derived for m-ary-input/symmetric-output channels, which gives a sufficient stability condition for Zm LDPC codes and will be complemented by a matched necessary stability condition introduced herein. Applications to the coded modulations and to codes with non-equiprobable distributed codewords will also be discussed. For binary codes, two new lower bounds are provided for symmetric channels, including a two-dimensional iterative bound and a one-dimensional non-iterative bound, the latter of which is the best known bound that is tight for BSCs. By adapting the reverse channel perspective, a pair of upper and lower bounds on the decodable Bhattacharyya noise parameter is derived for non-symmetric channels, which coincides with the existing bound for symmetric channels.