Long quasi-cyclic low-density parity check (QC-LDPC) codes have been chosen as one of the forward error correction (FEC) schemes in passive optical network (PON) systems benefiting from the excellent performance. About how to get the appropriate LDPC codes, some traditional methods which focus on obtaining large girth or flat approximated cycle extrinsic message degree (ACE) spectrums are proposed for the design of LDPC codes. These methods which may still make the existence of small stopping sets containing short cycles probably are not so useful for long QC-LDPC codes. Recently, the parallel vector message passing (PMP) algorithm oriented-to decreasing the number of short cycles is proved to be an effective method to optimize LDPC codes. Since ignoring short cycles with poor connectivity which can induce error floor, this method cannot guarantee its effectiveness for long QC-LDPC codes. In this letter, an edge exchange optimization method based on the global properties of ACE and girth called EEA method is proposed, which tries to reduce cycles with low ACE in the order of cycle length. The numerical results prove that our EEA method can bring the improvements of error floor up to two orders of magnitude to long QC-LDPC codes and obviously outperforms PMP algorithm.
Long quasi-cyclic low-density parity check (QC-LDPC) codes have been chosen as one of the forward error correction (FEC) schemes in passive optical network (PON) systems benefiting from the excellent performance. About how to get the appropriate LDPC codes, some traditional methods which focus on obtaining large girth or flat approximated cycle extrinsic message degree (ACE) spectrums are proposed for the design of LDPC codes. These methods which may still make the existence of small stopping sets containing short cycles probably are not so useful for long QC-LDPC codes. Recently, the parallel vector message passing (PMP) algorithm oriented-to decreasing the number of short cycles is proved to be an effective method to optimize LDPC codes. Since ignoring short cycles with poor connectivity which can induce error floor, this method cannot guarantee its effectiveness for long QC-LDPC codes. In this letter, an edge exchange optimization method based on the global properties of ACE and girth called EEA method is proposed, which tries to reduce cycles with low ACE in the order of cycle length. The numerical results prove that our EEA method can bring the improvements of error floor up to two orders of magnitude to long QC-LDPC codes and obviously outperforms PMP algorithm.
Many studies were carried out for the construction of low density parity-check (LDPC) codes. They usually focused on introducing the construction methods for good LDPC codes instead of a general method for code optimization. This paper proposes a method with high versatility, called the parallel vector message passing-based edge exchange (PMPE), for optimizing a type of graph-based LDPC codes, without changing the code parameters of mother codes, such as the code length, code rate, and degree distribution. With the approximately nearest codewords searching approach, we find the optimization method can increase the Hamming distance of the LDPC codes. For the quasi-cyclic (QC) LDPC codes, an optimization method, called the parallel vector message passing oriented-to the QC-LDPC codes (QC-PMP), is further suggested, with which the quasi-cyclic characteristics of QC-LDPC codes can remain unchanged in the optimization. To evaluate the performance of the parity-check matrix corresponding to a Tanner graph, a very simple metric, the cycles metric, is introduced to work with the proposed PMPE and QC-PMP algorithms. The experimental results show that the performance of the LDPC codes optimized with the proposed PMPE can be improved significantly at low BER range compared with the mother codes of the random codes, including the regular MacKay code of rate 0.5 and the regular PEG code of rate 0.9. For the case of the regular and irregular QC-LDPC codes with different code lengths and code rates, the optimized LDPC codes with the proposed QC-PMP algorithm significantly outperform the mother codes.
Recently, structured LDPC codes have been focused on due to excellent performance and lower complexity. An improved construction of QC-LDPC based on a modified PEG algorithm is proposed in this paper. The modified Progressive Edge-Growth algorithm is a PEG algorithm with Approximated Cycle Extrinsic Message Degree (ACE) metric, which is used to describe the connectivity of cycles. The approach can maximize the girth of the cycles, improve the connectivity of cycles and have the advantages of QC algorithms. The simulation results demonstrate that the PEG algorithm based on ACE has lower Bit Error Rate (BER) and Frame Error Rate (FER) than the original PEG algorithm at the low Signal Noise Ratio (SNR) values, and has a relatively lower computational complexity.
In this paper, we consider the construction of irregular QC-LDPC codes by jointly optimizing the girth, the number and approximate cycle extrinsic message degree (ACE) of short cycles to further improve the error-floor performance of irregular QC-LDPC codes with given degree distributions. By selecting the best edges, each of which must make the cycles consisting of the current variable node have the best lower bound of ACE or the minimum number of cycles with ACE lower bound, for each variable node via the progressive Edge growth (PEG) approaches, irregular QC-LDPC codes with better tradeoff between girth, the shortest cycle number and ACE could be constructed. Simulation results show that the proposed codes achieve better error-floor performance compared with the codes using the same degree distributions constructed by the recent proposed methods.
Gallager proposed and developed low density parity check (LDPC) codes in the early 1960s. LDPC codes were rediscovered in the early 1990s and shown to be capacity approaching over the additive white Gaussian noise (AWGN) channel. Subsequently, density evolution (DE) optimized symbol node degree distributions were used to significantly improve the decoding performance of short to medium length irregular LDPC codes. Currently, the short to medium length LDPC codes with the lowest error floor are DE optimized irregular LDPC codes constructed using progressive edge growth (PEG) algorithm modifications which are designed to increase the approximate cycle extrinsic message degrees (ACE) in the LDPC code graphs constructed. The aim of the present work is to find efficient means to improve on the error floor performance published for short to medium length irregular LDPC codes over AWGN channels in the literature. An efficient algorithm for determining the girth and ACE distributions in short to medium length LDPC code Tanner graphs has been proposed. A cyclic PEG (CPEG) algorithm which uses an edge connections sequence that results in LDPC codes with improved girth and ACE distributions is presented. LDPC codes with DE optimized/’good’ degree distributions which have larger minimum distances and stopping distances than previously published for LDPC codes of similar length and rate have been found. It is shown that increasing the minimum distance of LDPC codes lowers their error floor performance over AWGN channels; however, there are threshold minimum distances values above which there is no further lowering of the error floor performance. A minimum local girth (edge skipping) (MLG (ES)) PEG algorithm is presented; the algorithm controls the minimum local girth (global girth) connected in the Tanner graphs of LDPC codes constructed by forfeiting some edge connections. A technique for constructing optimal low correlated edge density (OED) LDPC codes based on modified DE optimized symbol node degree distributions and the MLG (ES) PEG algorithm modification is presented. OED rate-½ (n, k)=(512, 256) LDPC codes have been shown to have lower error floor over the AWGN channel than previously published for LDPC codes of similar length and rate. Similarly, consequent to an improved symbol node degree distribution, rate ½ (n, k)=(1024, 512) LDPC codes have been shown to have lower error floor over the AWGN channel than previously published for LDPC codes of similar length and rate. An improved BP/SPA (IBP/SPA) decoder, obtained by making two simple modifications to the standard BP/SPA decoder, has been shown to result in an unprecedented generalized improvement in the performance of short to medium length irregular LDPC codes under iterative message passing decoding. The superiority of the Slepian Wolf distributed source coding model over other distributed source coding models based on LDPC codes has been shown.
The construction of finite-length irregular LDPC codes with low error floors is currently an attractive research problem. In particular, for the binary erasure channel (BEC), the problem is to find the elements of selected irregular LDPC code ensembles with the size of their minimum stopping set being maximized. Due to the lack of analytical solutions to this problem, a simple but powerful heuristic design algorithm, the approximate cycle extrinsic message degree (ACE) constrained design algorithm, has recently been proposed. Building upon the ACE metric associated with a cycle in a code graph, we introduce the ACE spectrum of LDPC codes as a useful tool for evaluation of codes from selected irregular LDPC code ensembles. Using the ACE spectrum, we generalize the ACE constrained design algorithm, making it more flexible and efficient. We justify the ACE spectrum approach through examples and simulation results.
In this paper we introduce a novel progressive edge-growth (PEG) algorithm, the cyclic PEG (CPEG) algorithm. The CPEG algorithm uses an alternative edge establishment sequence to construct low-density parity-check (LDPC) codes. Irregular LDPC codes constructed using the CPEG algorithm have improved girth and approximate cycle extrinsic message degree (ACE) compared to existing PEG algorithms. We also analyze the minimum codeword weight, minimum stopping set weight, local girth, and local ACE distributions for codes in four very large ensembles of irregular LDPC codes. The code ensembles analyzed were constructed using standard PEG, ACE modified standard PEG, CPEG, and ACE modified CPEG algorithms. Modifications to improve the ACE in PEG LDPC codes, by Xiao and Banihashemi, were implemented in the ‘ACE modified’ versions of the PEG algorithms. The ACE modified standard PEG algorithm constructed the code ensemble with the highest minimum codeword weight and minimum stopping set weight distributions, and the ACE modified CPEG algorithm constructed the code ensemble with the highest local girth and ACE distributions. Short block length irregular LDPC codes with good degree distributions which have higher minimum weights than have been published for similar LDPC codes were found in the four code ensembles.