This paper describes a combinatorial approach to estimate the error rate performance of low-density parity-check (LDPC) codes decoded by (quantized) soft-decision iterative decoding algorithms. The method is based on efficient enumeration of input vectors with small distances to a reference vector whose elements are selected to be the most reliable values from the input alphabet. Several techniques, including modified cycle enumeration, and the efficient derivation of problematic inputs for finer quantizers from those of coarser ones are employed to reduce the complexity of the enumeration. The error rate estimate is derived by testing the input vectors of small distances followed by estimating the contribution of larger distance vectors. We demonstrate by a number of examples that the proposed method provides accurate estimates of error rate with computational complexity much lower than that of Monte Carlo simulations, especially at the error floor region.

The performance of low-density parity-check (LDPC) codes decoded by hard-decision iterative decoding algorithms can be accurately estimated if the weight J and the number |EJ| of the smallest error patterns that cannot be corrected by the decoder are known. To obtain J and |EJ|, one would need to perform the direct enumeration of error patterns with weight i les J. The complexity of enumeration increases exponentially with J, essentially as nJ, where n is the code block length. This limits the application of direct enumeration to codes with small n and J. In this letter, we approximate J and |EJ| by enumerating and testing the error patterns that are subsets of short cycles in the code's Tanner graph. This reduces the computational complexity by several orders of magnitude compared to direct enumeration, making it possible to estimate the error rates for almost any practical LDPC code. To obtain the error rate estimates, we propose an algorithm that progressively improves the estimates as larger cycles are enumerated. Through a number of examples, we demonstrate that the proposed method can accurately estimate both the bit error rate (BER) and the frame error rate (FER) of regular and irregular LDPC codes decoded by a variety of hard-decision iterative decoding algorithms.

The application of successive relaxation (SR) to the fixed-point problem associated with the iterative decoding of low-density parity-check (LDPC) codes was proposed by Hemati et al.. The simulation results presented by Hemati et al. for the SR version of belief propagation (BP) in the likelihood ratio (LR) domain and that of min-sum (MS) in the log-likelihood ratio (LLR) domain are based on the assumption of all-zero codeword transmission. This assumption however results in erroneous error rates when SR is applied in the LR domain. Here, we correct the simulation results reported by Hemati et al. for SR-BP in the LR domain. Furthermore, we investigate the performance of SR-BP and SR-MS in the LLR and LR domains, respectively. The results for a binary input additive white Gaussian noise (BIAWGN) channel show that for both BP and MS, the application of SR in the two domains of LR and LLR results in different error correcting performance. In particular, for the tested codes, it is shown that among the four algorithms, SR-MS-LLR has the best performance. It outperforms standard MS and BP by up to about 0.6 dB and 0.3 dB, respectively, offering an attractive solution in terms of performance/complexity tradeoff.

This paper describes a combinatorial approach to estimate the error rate performance of low-density parity-check (LDPC) codes decoded by (quantized) soft-decision iterative decoding algorithms. The method is based on efficient enumeration of input vectors with small distances to a reference vector whose elements are selected to be the most reliable values from the input alphabet. Several techniques, including modified cycle enumeration, are employed to reduce the complexity of the enumeration. The error rate estimate is derived by testing the input vectors of small distances and estimating the contribution of larger distance vectors. We demonstrate by a number of examples that the proposed method provides accurate estimates of error rate with computational complexity much lower than that of Monte Carlo simulations, especially at the error floor region.

The performance of low-density parity-check (LDPC) codes decoded by hard-decision iterative decoding algorithms can be accurately estimated if the weight J and the number |Ej| of the smallest error patterns that cannot be corrected by the decoder are known. To obtain J and |Ej|, one would need to perform the direct enumeration of error patterns with weight i les J. The complexity of enumeration increases exponentially with J, essentially as nJ, where n is the code block length. In this paper, we approximate J and |Ej| by enumerating and testing the error patterns that are subsets of short cycles in the code's Tanner graph. This reduces the computational complexity by several orders of magnitude compared to direct enumeration, making it possible to estimate the error rates for almost any practical LDPC code. To obtain the error rate estimates, we propose an algorithm that progressively improves the estimates as larger cycles are enumerated. Through a number of examples, we demonstrate that the proposed method can accurately estimate both the bit error rate (BER) and the frame error rate (FER) of regular and irregular LDPC codes decoded by a variety of hard-decision iterative decoding algorithms.

A method for estimating the performance of low-density parity-check (LDPC) codes decoded by hard-decision iterative decoding algorithms on binary symmetric channels (BSCs) is proposed. Based on the enumeration of the smallest weight error patterns that cannot be all corrected by the decoder, this method estimates both the frame error rate (FER) and the bit error rate (BER) of a given LDPC code with very good precision for all crossover probabilities of practical interest. Through a number of examples, we show that the proposed method can be effectively applied to both regular and irregular LDPC codes and to a variety of hard-decision iterative decoding algorithms. Compared with the conventional Monte Carlo simulation, the proposed method has a much smaller computational complexity, particularly for lower error rates.

The progressive-edge-growth (PEG) algorithm (X.-Y. Hu et al., 2001; 2002) is known to construct low-density parity-check codes at finite block lengths with very good performance. We propose a very simple modification to the PEG construction for irregular codes, which considerably improves the performance at high signal-to-noise ratios (SNR) with no sacrifice in low-SNR performance.

We study a wide range of graph-based message-passing schedules for iterative decoding of low-density parity-check (LDPC) codes. Using the Tanner graph (TG) of the code and for different nodes and edges of the graph, we relate the first iteration in which the corresponding messages deviate from their optimal value (corresponding to a cycle-free graph) to the girths and the lengths of the shortest closed walks in the graph. Using this result, we propose schedules, which are designed based on the distribution of girths and closed walks in the TG of the code, and categorize them as node based versus edge based, unidirectional versus bidirectional, and deterministic versus probabilistic. These schedules, in some cases, outperform the previously known schedules, and in other cases, provide less complex alternatives with more or less the same performance. The performance/complexity tradeoff and the best choice of schedule appear to depend not only on the girth and closed-walk distributions of the TG, but also on the iterative decoding algorithm and channel characteristics. We examine the application of schedules to belief propagation (sum-product) over additive white Gaussian noise (AWGN) and Rayleigh fading channels, min-sum (max-sum) over an AWGN channel, and Gallager's algorithm A over a binary symmetric channel.