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IEEE Transactions on Communications

Set Message-Passing Decoding Algorithms for Regular Non-Binary LDPC Codes

Qin Huang, Liyuan Song, Zulin Wang

Published Aug 28, 2017• 19 citations
Publisher

Abstract

In the check node (CN) update of non-binary message-passing algorithms, each element of reliability vectors takes the same computational complexity. However, our analysis indicates that various elements in the same vector have various correct probabilities, thus have different contributions to error performance. In order to match computational complexity with correct probability, all elements in a vector are partitioned into different sets. For the extended min-sum (EMS) decoding, various strategies are applied for sets according to their correct probability. For the trellis-based EMS decoding, it is interesting that set partition only involves fixed paths, thus it does not need to search over the whole trellis of a CN. Complexity analysis and simulation results show that the proposed algorithms efficiently decode non-binary low-density parity-check codes, including ultra-sparse ones.

DOI
10.1109/tcomm.2017.2746101

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In the check node (CN) update of non-binary message-passing algorithms, each element of reliability vectors takes the same computational complexity. However, our analysis indicates that various elements in the same vector have various correct probabilities, thus have different contributions to error performance. In order to match computational complexity with correct probability, all elements in a vector are partitioned into different sets. For the extended min-sum (EMS) decoding, various strategies are applied for sets according to their correct probability. For the trellis-based EMS decoding, it is interesting that set partition only involves fixed paths, thus it does not need to search over the whole trellis of a CN. Complexity analysis and simulation results show that the proposed algorithms efficiently decode non-binary low-density parity-check codes, including ultra-sparse ones.
This paper reduces the complexity of decoding non-binary low-density parity-check (LDPC) codes by set partition. In the check node update, the input vectors are partitioned into several sets such that different elements in the virtual matrix enjoy various computational strategies. As a result, the proposed algorithm achieves high computational efficiency by setting strategies according to the correct probability of these elements. Simulation results indicate that it significantly decreases the complexity of check node update with negligible performance loss.
In this paper, we present an improvement and a new implementation of a simplified decoding algorithm for non-binary low density parity-check codes (NB-LDPC) in Galois fields GF(q). The base algorithm that we use is the Extended Min-Sum (EMS) algorithm, which has been widely studied in the recent literature, and has been shown to approach the performance of the belief propagation (BP) algorithm, with limited complexity. In our work, we propose a new way to compute modified configuration sets, using a trellis representation of incoming messages to check nodes. We call our modification of the EMS algorithm trellis-EMS (T-EMS). In the T-EMS, the algorithm operates directly on the deviation space by considering a trellis built from differential messages, which serves as a new reliability measure to sort the configurations. We show that this new trellis representation reduces the computational complexity, without any performance degradation. In addition, we show that our modifications of the algorithm allows to greatly reduce the decoding latency, by using a larger degree of hardware parallelization.
Low Density Parity Check (LDPC) codes have been widely applied in recent years, and Extended Min-Sum (EMS) decoding algorithm is the most practical algorithm, which has a fairly low computational cost. Due to the great loss in error correction performance of EMS algorithm, in this paper, we establish constraints to obtain reliable nodes while decoding and evaluate the reliability of the nodes. The nodes and information transferred between nodes are processed and optimized according to the reliability of the nodes. High reliability nodes will not be updated in the decoding process while the information of the sub-reliability nodes will be optimized in different situations, thereby improving the decoding accuracy. This algorithm can accelerate the convergence of the algorithm, while the reduced number of update nodes can control the increase of the computation to a certain extent, and maintain the advantage of the low computation quantity of EMS algorithm. Through simulation results and computational analysis, we prove that the improved algorithm has significantly improved error correction performance compared with the traditional EMS algorithm. Under the same conditions, it has completely entered the waterfall area at 3.6dB, and the EMS algorithm has not entered the waterfall area. By comparing the average iteration times of decoding of the two algorithms, the convergence speed of the algorithm can be reflected. Compared with EMS algorithm, the improved algorithm has significantly improved the convergence speed in most areas, which indicates that the processing of sub-reliable nodes accelerates the convergence of the algorithm. Finally, the average update times of the check nodes and variable nodes of the two algorithms are compared respectively. Although the improved algorithm introduces extra processing flow, it speeds up the convergence of the algorithm, and the computational cost of decoding is less expensive.
This paper addresses binary message-passing (MP) decoding for nonbinary low-density parity-check (NB-LDPC) codes based on the binary image of Galois field symbols. The parity-check matrix of NB-LDPC codes in binary form is used to perform MP. The corresponding nonbinary check node (CN) update and variable node (VN) update can thus be decomposed to a set of binary sub-CN updates and sub-VN updates with much lower computational complexity. In particular, we start from adapting the binary parity-check matrix with Gaussian elimination. Then, we add redundant rows to the parity-check matrix instead of adaptation to further improve the performance. A min-max operation based on the expanded matrix is proposed for the CN update. It not only decreases the computational complexity incurred by Gaussian elimination but also improves the error performance. Simulation results show that the bit-level decoding with min-max operation can achieve similar error performance as the extended min-sum algorithm, but the computational complexity can be lower than the min-sum algorithm for binary LDPC codes.
The fixed path Min-Sum (FMS) algorithm efficiently decodes non-binary LDPC codes with low/moderate code rates, but suffers significant performance degradation for high code rates. This paper enhances its check node (CN) update by selecting deviation paths of high reliability according to their correct probabilities and positions over the trellis of a CN. Moreover, a path pruning strategy is introduced during the CN update to reduce the complexity of path search. Analysis results indicate that the proposed pruning path Min-Sum (PMS) algorithm efficiently decodes high-rate non-binary LDPC codes.
During the check node (CN) update, the elements of input message vectors are redundant for the output message vectors. Hence, in this paper, we exactly select from the input message vectors the elements, which really have contributions to the error-correction performance and constitute the minimum set for the CN update. With adoption of the forward and backward (FB) scheme, an adaptive minimum-set min-sum algorithm (AMSA) is proposed to reduce the computation complexity of the FB process. In order to concurrently update the output vectors belonging to the same CN, we present a parallel minimum-set min-sum algorithm (PMSA) with lower memory complexity than the AMSA. Compared with the min-sum algorithms, the two proposed minimum-set based algorithms introduce no error performance loss.