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JOURNAL OF ELECTRONICS INFORMATION TECHNOLOGY

Two Low-complexity Symbol Flipping Decoding Algorithms for Non-binary LDPC Codes

Haiqiang Chen, Yaolin Wang, Wenjuan Wei, Bingxu Jiang, Youming Sun, Xiangcheng Li, Tuanfa Qin

Published Jan 15, 2021• 0 citations
Publisher

Abstract

Two low-complexity symbol flipping decoding algorithms, the Improved weighted-Algorithm B algorithm (Iwtd-AlgB) and the Truncation-based Distance-Symbol-Flipping-Decoding with Prediction (TD-SFDP) algorithm, are presented for non-binary Low Density Parity Check (LDPC) codes. For the Iwtd-AlgB algorithm, the scaling factor of the flipping metric can be replaced by the simple sums of the extrinsic information and the distance-based parameter, which can avoid the multiplication operations in the iterations and thus can reduce the decoding complexity. For the presented TD-SFDP algorithm, the variable nodes and the finite field symbols are truncated and classified based on the extrinsic information frequency and the flipping function. Only those nodes/symbols that satisfy the designed conditions can be involved in the message updating process. Simulations and numeric results show that, the presented two decoding algorithms can reduce the computational complexity at each iteration with a controllable performance degradation, thus can make efficient trade-offs between performance and complexity.

DOI
10.11999/jeit191008

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Two low-complexity symbol flipping decoding algorithms, the Improved weighted-Algorithm B algorithm (Iwtd-AlgB) and the Truncation-based Distance-Symbol-Flipping-Decoding with Prediction (TD-SFDP) algorithm, are presented for non-binary Low Density Parity Check (LDPC) codes. For the Iwtd-AlgB algorithm, the scaling factor of the flipping metric can be replaced by the simple sums of the extrinsic information and the distance-based parameter, which can avoid the multiplication operations in the iterations and thus can reduce the decoding complexity. For the presented TD-SFDP algorithm, the variable nodes and the finite field symbols are truncated and classified based on the extrinsic information frequency and the flipping function. Only those nodes/symbols that satisfy the designed conditions can be involved in the message updating process. Simulations and numeric results show that, the presented two decoding algorithms can reduce the computational complexity at each iteration with a controllable performance degradation, thus can make efficient trade-offs between performance and complexity.
The symbol flipping decoding algorithms based on prediction (SFDP) for non-binary LDPC codes perform well in terms of error performances but converge slowly when compared to other symbol flipping decoding algorithms. In order to improve the convergence rate, we design new flipping rules with two phases for the SFDP algorithms. In the first phase, two or more symbols are flipped at each iteration to allow a quick increase of the objective function. While in the second phase, only one symbol is flipped to avoid the oscillation of the decoder when the objective function is close to its maximum. Simulation results show that the SFDP algorithms with the proposed flipping rules can reduce the average number of iterations significantly, whereas having similar performances when compared to the original SFDP algorithms.
Symbol flipping decoding (SFD) of non-binary low-density parity-check (NB-LDPC) codes has lower implementation complexity at a cost of performance degradation, when compared with other belief propagation decoding algorithms. This paper proposes modified SFD algorithms based on prediction (SFDP) for NB-LDPC codes by introducing random penalty items into the flipping metrics. Two types of noise with uniform and Gaussian distributions are considered. A probabilistic analysis shows theoretically that the randomly penalized SFDP algorithms can be advantageous in the correct flipping probability over their original versions. Simulation results show that the proposed penalizing methods exhibit significant performance gains for both regular and irregular NB-LDPC codes with different rates and alphabet sizes in both the additive white Gaussian noise (AWGN) channel and the binary symmetric channel (BSC).
Based on the symbol-flipping algorithm with multiple-votes (MV-SF), this paper presents two simplified algorithms, symbol flipping with truncated and multiple votes (T-MV-SF) and symbol flipping with simplified and multiple votes (S-MV-SF), for nonbinary low-density parity-check(NB-LDPC) codes. Unlike previous symbol-flipping based algorithms (SFBAs), the T(S)-MV-SF uses the truncated voting information in each check and variable node update. The proposed algorithms also approximate the sorting function in each check update. Therefore, their complexity and memory consumption are remarkably reduced as compared with existing SFBAs. Furthermore, we introduce several parameters to compensate the information loss caused by the truncated voting information. After optimizing all the parameters, the proposed algorithms even have better performance than the MV-SF. As compared with the MV-SF, the maximum performance gain among all numerical examples is about 0.11 dB for the S-MV-SF on the (63, 37) code. The T-MV-SF has the largest reduction of the computational complexity on the (255, 175) code, which is about 80% of the complexity of the MV-SF. Moreover, the performance of the T-MV-SF is better than the MV-SF on the (255, 175) code. As a result, the proposed algorithms can effectively balance the error performance and complexity for decoding of NB-LDPCs.
In this paper, we present two low complexity algorithms to decode non-binary LDPC codes. The proposed decoding algorithms update iteratively the hard decision received vector to search for a valid codeword in the vector space of Galois field (GF). The selection criterion for the position of unreliable symbols is based on failed checks and the information from the Galois field structure. In the first proposed algorithm, the flipping function is calculated for all symbols of the received sequence and multiple symbols are flipped in each iteration while in the second proposed algorithm, a single symbol is flipped per iteration. In the second method, unreliable positions are short-listed by using a majority voting scheme, and then the flipping function is computed to predict candidate symbols from the set of symbols in GF(q) while not violating the field order q. The proposed methods reduce the decoding complexity and memory use. The results of the algorithms show appealing tradeoffs between complexity and bit error rate performance for non-binary LDPC codes.
A properly designed stopping criterion for iterative decoding algorithms can save a number of iterations and lead to a considerable reduction of system latency. The symbol flipping decoding algorithms based on prediction (SFDP) have been proposed recently for efficient decoding of non-binary low-density parity-check (LDPC) codes. To detect the decoding frames with slow convergence or even non-convergence, we track the number of oscillations on the value of objective function during the iterations. Based on this tracking number, we design a simple stopping criterion for the SFDP algorithms. Simulation results show that the proposed stopping criterion can significantly reduce the number of iterations at low signal-to-noise ratio regions with slight error performance degradation.
A hybrid decoding algorithm which proposed for nonbinary and binary low density parity (LDPC) codes, both the algorithm combines the weighted symbol flipping (WSF) algorithm with the fast Fourier transform q-ary sum product algorithm (FFT-QSPA).The flipped position and value are determined by the symbol flipping metric and the received bit values in the first stage WSF algorithm. If the low complexity WSF algorithm is failed, the second stage FFT-QSPA is activated as a switching strategy. They are particularly effective for decoding LDPC codes constructed based on finite geometries and finite fields. Analyzing both the techniques nonbinary LDPC codes gives the better error performance and greatly reduces the computation complexity compared to binary LDPC codes. The proposed hybrid algorithm is used for some applications in communication systems for high speed and low power consumption. Keywords Nonbinary and binary low density parity-check (LDPC) code, Weighted symbol –flipping (WSF), Hybrid weighted symbol-flipping (HWSF), iterative decoding. 1.