An optimized initialization for LDPC decoding over GF(q) in impulsive noise environments.

Modern navigation satellite communication has the characteristic of high transmitting rate. To avoid bit errors in data transmission, low density parity check (LDPC) codes are widely recognized as efficient ways for navigation communication. Conventionally, the LDPC decoding is applied for additive white Gaussian noise (AWGN) channel and degrades severely while facing the impulsive noise. However, navigation communication often suffers from impulsive interference due to the occurrence of high amplitude "spikes". At this time, the conventional Gaussian noise assumption is inadequate. The impulsive component of interference has been found to be significant which influences the reliability of transmitted information. Therefore the LDPC decoding algorithms for AWGN channel are not suitable for impulsive noise environments. Consider that LDPC codes over GF(q) perform better than binary LDPC in resisting burst errors for current navigation system, it is necessary to conduct research on LDPC codes over GF(q). In this paper, an optimized initialization by calculating posterior probabilities of received symbols is proposed for non-binary LDPC decoding on additive white Class A noise (AWAN) channel. To verify the performance of the proposed initialization, extensive experiments are performed in terms of convergence, validity, and robustness. Preliminary results demonstrate that the decoding algorithm with the optimized initialization for non-binary LDPC codes performs better than the competing methods and that of binary LDPC codes on AWAN channel.

Introduction

With the explosive development of communication technology, the new mobile communication systems, such as beyond fifth generation (B5G) and sixth generation (6G) systems, will suffer from severe challenges imposed by the requirement for heavy connection density and high efficiency [1]. Especially for 6G, satellite communications play an important role in providing high quality communication services to achieve the worldwide connectivity [2]. As one of the critical components, modern navigation satellite communication has the characteristic of high transmitting rate which is a real challenge to ensure the correctness of transmitted information in various channels [3,4]. To avoid bit errors in data transmission, it is an efficient way to employ low density parity check (LDPC) codes in navigation satellite communication channel. Nowadays, LDPC codes have been widely applied in the 2nd-generation digital video broadcast via satellite (DVB-S2) standard [5], 5G mobile communications [6], DNA barcoding [7], as well as the uplink of mobile satellite communication [8].

For modern navigation system, LDPC codes over GF(q) have been employed in encoding and decoding as important codes. With the rapid development of BeiDou Navigation Satellite System (BDS) 3, B1C signal and B2a signal have been utilized gradually. The B-CNAV1 navigation message is transmitted through B1C signal. Each frame of message consists of 3 sub frames, of which the second sub frame exploits LDPC codes (200,100) over GF(64) and the third sub frame leverages LDPC codes (88, 44) over GF(64) for encoding [9]. Meanwhile, the B-CNAV2 navigation message is transmitted through B2a signal with LDPC codes (96, 48) over GF(64). Non-binary LDPC codes have been recognized as a powerful technology to encode navigation information efficiently. With the continuous development of BDS 3 and the increasing demand for navigation, people hope to access precise navigation information in time. However, working in severe environments, such as underwater environment surrounded by acoustical noises or in the power industry, the decoder of non-binary LDPC codes is inevitably subject to impulsive interference, which causes poor reliability of transmitted information and even leads to communication failure. Therefore, nowadays, evaluation and analysis of LDPC codes over GF(q) for navigation satellite communication, especially in harsh environments, arouse attentions of researchers all over the world.

Conventionally, the LDPC decoding algorithms are designed for AWGN channel. However, navigation satellite communication often suffers from irregular noises due to the occurrence of high amplitude “spikes”. For navigation satellite communication, such spikes can be generated in atmosphere where lightning discharges in the vicinity of the receiver, or underwater environment where the ambient acoustical noises includes impulses due to noisy aquatic animals such as snapping shrimp [10-13]. At this time, the original assumption that noises are the Gaussian noises is inadequate. Unfortunately, few attention has been given to decoding algorithms for LDPC codes over GF(q) in impulsive noise environments.

Generally, compared with binary LDPC, LDPC codes over GF(q) show superior performance in resisting burst errors such as impulsive noises for the characteristic of inner interleaving [14,15], which makes non-binary LDPC codes more suitable for navigation communication. Furthermore, non-binary LDPC codes combined with q-ary modulation can increase transmission rate obviously [16,17]. Although there is an increment of computational complexity by adopting LDPC codes over GF(q), with the development of terminal computation, non-binary LDPC codes have become a hot research topic and would become more prevalent and applicable. Therefore, we focus on non-binary LDPC codes decoding on additive white Class A noise (AWAN) channel and present an optimized initialization for decoding in this paper. With simulation experiments, we demonstrate the efficiency of the proposed algorithm. The main contributions are summarized as follows:

We investigate the problem of LDPC decoding in impulsive noise environments for navigation communication and formalize the impulsive noise as the Class A noise model.

We propose an optimized initialization by calculating posterior probabilities of received symbols for non-binary LDPC decoding on AWAN channel, which makes use of series truncation for computing effectively.

Extensive experiments are conducted on convergence, validity, and robustness. The experimental results reveal that the optimized initialization has a significant effect on the decoding performance for non-binary LDPC codes on AWAN channel.

The rest of the paper is organized as follows. We illustrate the related work in Section 2. Section 3describes AWAN model based on the statistics and conventional initialization of decoder for LDPC codes over GF(q). Section 4illustrates the optimized initialization process on AWAN channel. Simulation results are shown and discussed in Section 5. Finally, the conclusion is given in Section 6.

Related work

LDPC codes were introduced by Gallager in 1962 for the first time, which are linear codes with sparse parity-check matrix [18]. In 1996, Mackay and Wiberg found that LDPC codes achieved excellent performance approximating the Shannon limit, and it soon became a research hotspot in the channel coding theory [19]. To improve the performance of error correction and transmission, Davey and MacKay extended the belief-propagation (BP) decoding algorithm for binary LDPC to non-binary LDPC firstly [20]. Furthermore, they proposed a fast Fourier Transform-based belief-propagation (FFT BP) decoding algorithm for reducing complexity [21]. Unfortunately, this method was only valid when the Galois field was a binary extension field with q = 2 and inefficient to handle other situations. To further simplify the decoding procedure, Declercq et al. introduced an extended min-sum algorithm, but it led to degradation of performance [14]. The United States adopted LDPC codes with 1/2 code rate in the L1C signal of GPS for the first time. Hareedy Ahmed et al. proposed non-binary LDPC codes for magnetic recording channels, and provided a comprehensive analysis of the error floor along with codes optimization guidelines for structured and regular non-binary LDPC codes [22]. In 2017, Huang Qin proposed message-passing decoding algorithms that decoded non-binary LDPC codes including ultra-sparse ones efficiently [23]. In 2019, Rehman employed parallel architecture for LDPC codes decoding to achieve the higher data rate, which in turn raised the memory conflict issue [24]. In recent years, with the emergence of decoding algorithms with low complexity, LDPC codes show superiority in practice with their excellent capability, and have been replacing the conventional codes as main codes in future navigation satellite communication gradually.

Generally, receivers adopts parameters of the AWGN channel and conventional decoding methods for LDPC decoding, which results in serious degradation in impulsive noise environments. The impulsive component of interference has been found to be significant which influences the reliability of transmitted information.

Various attempts have been made to develop models of impulsive noises that can be divided into empirical models and physical models. Class A noise model proposed by Middleton is a typical kind of physical models [25]. The statistical feature of Class A noise is much different from that of Gaussian noise, therefore the LDPC decoding algorithms for AWGN channel are not suitable for Class A noise environments. Maad et al. analyzed the performance of LDPC codes in heavy-tailed, symmetric alpha stable noise (SαS) channels [26]. Further, Nakagawa et al. proposed the sum-product decoding method for binary LDPC codes in Class A noise environment [27]. However, few researches have ever explored decoding algorithms for LDPC codes over GF(q) in Class A noise environments.

This paper is focused on the design of an optimized initialization for non-binary LDPC codes on AWAN channel. In contrast to metaheuristics, our optimized initialization is dedicated to deal with decoding problems based on BP. Generally, as the famous optimization techniques, metaheuristics are widely recognized as efficient approaches for optimization problems, such as particle swarm optimization (PSO) [28,29] and differential evolution (DE) [30]. Various metaheuristic methods have reported advantages in image segmentation [31], tuning hyper-parameters of deep neural networks [32,33], and benzene prediction model [34]. However, as illustrated in [35], the successful application of metaheuristics requires to find a good initial parameter setting, which is a tedious and time consuming task. Moreover, the performance of metaheuristics deteriorates quickly as the dimensionality increases, nevertheless high-dimensional circumstances are extremely common in encoding and decoding. Therefore, decoding algorithms based on BP are considered for non-binary LDPC codes on AWAN channel, and an optimized initialization by calculating posterior probabilities of received symbols is proposed.

Theoretical background

AWAN model

When encountering lightning or water during transmission, navigation signals will be interfered by the impulse noise, which leads to the unexpected change of the amplitude of the transmitted signal. In this part, the Class A noise model devised by Middleton [25] is described as a statistical AWAN model with the impulsive noise environment, which is widely applicable by adjusting parameters and provides fine closeness to experimental values. According to this theory, the Class A noise model is composed of Gaussian noise G(t) and impulsive noise X(t), which can be expressed as where G(t) is considered as background noise and X(t) can be expressed as [25]

Here, U denotes the j-th received impulse noise waveform from an interfering source and ϑ represents the random parameters which describe the waveform scale and structure. Assume that there is only one type of waveform, and U is generated with appropriate variations in the individual wave form under the variation of the parameter ϑ. According to the Class A noise model, the probability density function (PDF) of the noise amplitude z can be defined as where , A is the impulsive coefficient which is defined as average number of impulses on the receiver in unit time. is the Gaussian-to-Impulsive noise power ratio (GIR) with Gaussian noise power and impulsive noise power . Therefore the total noise power is .

The Class A noise in (3) consists of the impulsive noise with variance and the background Gaussian noise with variance . The number of impulsive noise is distributed with Poisson distribution (e-∙A)/m! and the amplitude of each impulsive noise is characterized by a Gaussian PDF with variance . Therefore, at a certain observation time, assume that the number of impulsive noise is m, which is characterized by a Poisson distribution with mean A, the noise of receiver is characterized by a Gaussian PDF with variance .

Consider the independence between the background Gaussian noise and the impulsive noise, as the impulse coefficient A increases, the impulsive noise becomes more intensive and continuous, which makes the Class A noise approximate the Gaussian noise. In particular, if A is close to 10, the statistical feature of the Class A noise is almost similar to that of the Gaussian noise [25]. In addition, as Γ grows, i.e., the proportion of Gaussian white noise in the total noise increases, the Class A noise gets close to the Gaussian noise. Otherwise, the smaller Γ is, the more impulsive the Class A noise would be.

Conventional initialization in decoding process

Non-binary LDPC codes can be considered as a kind of linear block codes which are the extension of binary LDPC codes over GF(q). The difference between non-binary LDPC codes and binary LDPC codes is that each non-zero element of sparse parity-check matrix needs to be obtained from GF(q). The Tanner graph of non-binary LDPC codes given by a sparse parity-check matrix over GF(q) is constructed in the same way as that of binary LDPC codes. Compared with binary LDPC codes, non-binary LDPC codes perform better in communication due to advantages in resisting burst error and high transmission rate [36-39].

A significant amount of research has been concentrated on the design, encoding, decoding and performance analysis of non-binary LDPC codes. Iterative decoding algorithm based on BP is an important soft decision decoding algorithm. According to this algorithm, messages are delivered between variable nodes and check nodes during iterations after initialization. Further, the received codes are updated until satisfying the parity-check equations or the upper limit of the iteration number is reached.

Since the channel transition probability is only utilized in the initialization process, we focus on optimizing initialization of LDPC decoding on AWAN channel. In this subsection, traditional initialization method for non-binary decoding is introduced. Consider that the codeword u = (u0,u1,…,u) is obtained by encoding information sequence e = (e0,e1,…,e). Q-ary sequence u can be expanded into binary sequence ((u0,1,u0,2,…,u0,),…,(u,u,…,u)) and modulated by binary phase shift keying (BPSK) (using the mapping 0 to 1, 1 to -1) to obtain the transmitted sequence x = ((x0,1,x0,2,…,x0,),…,(x,x,…,x)), where p = log2q. And y = ((y0,1,y0,2,…,y0,),…,(y,y,…,y)) is the received sequence transmitted on AWGN channel with variance σ2 and mean 0.

The initial messages sent from variable node v to the check node c is the probabilities of the j-th code symbol u equal to {a1,a2,…,a} respectively, given received sequence y. And it can be expressed as where 1≤k≤q and a denotes the element of GF(q). We expand a to binary sequence {a,a,…,a} and the Eq (4) can be rewritten as:

There is a mapping between expanded codeword u and transmitted sequence x due to BPSK modulation. Thus, P(u = a|y) equals to P(x = 1−2a|y). According to the Bayes formula, the posterior probabilities of received bits can be acquired by where a= ±1, P(x = ±a) = 0.5 and P(∙) is the Gaussian PDF with mean 0 and variance . Due to thus

And the initial messages of BP algorithm for LDPC codes over GF(q) could be obtained by adding (8) to (5).

An optimized initialization for LDPC decoding over GF(q) on AWAN channel

The conventional decoding methods degrade severely while facing the impulsive noise since those methods acquire the posterior probability of received symbols by making use of the transition probability on AWGN channel. To tackle this problem, based on the BP decoding algorithm, we present an optimized initialization for LDPC decoding over GF(q) on AWAN channel with series truncation.

As mentioned above, the initialization is the crucial part of the BP decoding over GF(q) algorithm. The posterior probability of received symbols is considered as the initial message transmitted from the variable nodes to the check nodes. Thus, to evaluate the posterior probability of received symbols correctly is vital for decoding.

During the initialization of conventional decoding process, we obtain the posterior probability of the received bits from the transition probability of the AWGN channel as Eq (9), which suffers from degradation of the bit error rate (BER) in impulsive noise environments. Therefore, for LDPC decoding over GF(q) on AWAN channel, since the channel parameters are only leveraged in the decoding initialization, we need to improve the initialization of the iterative decoding process without changing the information transmission mode between variable nodes and check nodes. Under this circumstance, the optimized initialization can be applied to various traditional LDPC decoding algorithms over GF(q), such as FFT BP and EMS algorithm. Generally, the BP decoding algorithm with optimized initialization for LDPC codes over GF(q) on AWAN channel can be illustrated by the flow chart in Fig 1.

Fig 1

The flow chart of BP decoding for LDPC codes over GF(q) on AWAN channel.

In the following part, we elaborate the optimized initialization in FFT BP decoding algorithm and series truncation is proposed to calculate the PDF of the Middleton Class A noise effectively. The LLR of the received bit can be defined as where 0≤jBPSK modulated bits.

FFT BP decoding with the proposed initialization (i.e. the optimized FFT BP algorithm) for non-binary LDPC codes on AWAN channel is illustrated in Algorithm 1, where t denotes the t-th iteration, E(u) is the layered message being u (u∈GF(q)) and initialized with the channel message Pn(u) in Eq (5), represents the message from a check node (CN) to a variable node (VN), indicates the message from VNn to CNm, and k is a normalizing constant to ensure is the set of variable nodes connected to CN, and φ(m)/n is the set of variable nodes connected to CN except VN. φ(n) is the set of check nodes connected to VN, and φ(n)/m is the set of check nodes connected to VN except CN. P(∙) and P-1(∙) denote the permutation function and inverse permutation function [40]. F(∙) and F-1(∙) indicate the FFT function and IFFT function. denotes the permutation message of . is the Fourier domain message of . is the check node updated message in the probability domain. indicates the check node updated message in the Fourier domain. also indicates the inverse permutation message of .

In practice, we need to estimate the impulsive coefficient A, GIR Γ and the background Gaussian noise power properly. Fortunately, these parameters can be estimated from the second, fourth and sixth epochs of the received envelopes [41].

Furthermore, since the PDF of the Class A noise PA(z) in (3) consists of infinite series, the calculation of LLR is extremely complex practically. Thus, the series truncation can be regarded as a valid alternative to compute effectively by

To obtain an appropriate L, we need to analyze the PDF of the Class A noise. Given the noise amplitude z, the value of each term in (14) is mainly influenced by impulsive coefficient A and equal to the product of Poisson component and Gaussian component. Fig 2 illustrates the relationship between Poisson component and term number m with different A. From Fig 2, we can observe that for the smaller A, the term m corresponding to the maximum value of Poisson probability becomes smaller. If A = 1, Poisson probability is close to 0 under the condition of m≥4, which leads to a negligible value for the m-th term of PA(z). Thus, L is considered as the minimal integer whose Poisson cumulative distribution with mean A is more than 99%:

Fig 2

Poisson probability vs. the term number m with different mean A.

According to (15), L equals to 3 with A = 1. PA(z) with different L is shown in Fig 3 and we can find that the curves are overlapped with L>3, which indicates formula (15) is effective.

Fig 3

PDF of noise amplitude z for A = 1 and L = 0, 1, 2, ….

After introducing the optimized BP decoding of non-binary LDPC codes on AWAN channel, we investigate the characteristics of LLR on AWGN channel and AWAN channel with A = 0.1, Γ = 0.1. Suppose that the code rate is 1/2 and the signal-to-noise ratio E/N0 is 0 dB for the received signal, where E denotes the energy per information bit and N0 is the one-sided Gaussian noise power spectral density. According to the relationship between E/N0 and the background Gaussian noise power , the following formulas can be acquired:

Fig 4 depicts the characteristics of LLR with different amplitudes of the received signal on AWGN and AWAN channels. From Fig 4, we can draw the conclusion that, the LLR is proportional to the amplitude of the received signal for AWGN channel; while for AWAN channel, it has a nonlinear relationship with the amplitude of the received signal. Specifically, for AWAN channel, the LLR is proportional to the amplitude of the received signal and similar to that of AWGN channel when the signal amplitude is less than 1. On the other hand, as the received signal amplitude increases, the corresponding LLR degrades due to that the received signal may contain impulsive noise with high probability.

Fig 4

LLR for AWGN and AWAN channels.

Results and discussion

In this section, series of experiments are conducted to demonstrate the efficient performance of the proposed optimized initialization for LDPC decoding over GF(q) on AWAN channel. Irregular quasi-cyclic (QC) LDPC codes (88, 44) and (200, 100) over GF(64) with bit rate of 1/2 and a binary irregular QC-LDPC code (528, 264) with bit rate of 1/2 were utilized in our experiments. And the decoder was set to perform at most 50 iterations. Different decoding methods are evaluated and compared with respect to convergence, validity, and robustness. Experimental results indicate the superiority of the proposed optimized initialization.

Convergence comparison

Firstly, convergence issues have been considered as in [42] and performance of different initializations was evaluated utilizing the BER. To be specific, we compared the BER of the FFT BP decoding algorithm with the optimized initialization (i.e. the optimized FFT BP algorithm) with that of the conventional FFT BP algorithm (designed for AWGN channel) on AWAN channel. We should note that the difference between the two algorithms is the initialization. Therefore, the statistical evaluation was performed using the Wilcoxon rank test metric of both algorithms for QC LDPC codes (88, 44) with the statistical significance value α = 0.05. The null hypothesis H0 is ‘The difference between E/N0 obtained by the optimized FFT BP algorithm and the conventional FFT BP algorithm is identical with the same BER’. Meanwhile, the alternative hypothesis is set as ‘the optimized FFT BP algorithm is validated’.

Table 1 presents the convergence comparisons of different initializations using the Wilcoxon Signed-Rank Test metric at different BER, where the ‘+’ indicates the cases when the algorithm acquires better coding gain. It clearly shows that the FFT BP algorithm with the proposed optimized initialization is statistically more superior.

Table 1

Results of convergence comparisons at different BER by utilizing Wilcoxon rank test (α = 0.05).

BER	10−1	10−2	10−3	10−4	10−5
p value	0	0	0	0	0
The optimized FFT BP	+	+	+	+	+

More specifically, as shown in Fig 5, the BER performance of the conventional FFT BP algorithm for the QC LDPC code (88, 44) suffers from large degradation on AWAN channel with A = 0.1 and Γ = 0.1. Actually, in contrast to the BER on AWGN channel, the conventional FFT BP on AWAN channel suffers about 14 dB degradation at BER = 10−5 because of the emergence of the impulse noise. In particular, assume that the GIR , then the total noise power σ2 of AWAN channel is 11 times of . If the total noise power on AWAN channel is the same as that on AWGN channel, the performance of the conventional FFT BP on AWAN channel suffers about 3.6 dB degradation than that on AWGN channel at BER = 10−5.

Fig 5

BER performance of the optimized FFT BP algorithm on AWAN channel.

Further, from Fig 5, we can find that the proposed optimized FFT BP decoding algorithm with series truncation performs efficiently on AWAN channel. More precisely, it achieves about 12.2 dB coding gain at BER = 10−5 compared to the conventional one.

Validity analysis

Secondly, the BER performance of LDPC code (88, 44) over GF(64) was compared with the binary LDPC code (528,264) on AWAN channel with A = 0.1, Γ = 0.1 in Fig 6. The decoding algorithm of binary LDPC codes on AWAN channel was introduced by Nakagawa et al. in [27]. Fig 6 demonstrates that the BER performance of binary LDPC codes on AWAN channel suffers from about 1.7 dB degradation than LDPC codes over GF(64), which reveals that LDPC codes over GF(64) outperform binary LDPC codes on AWAN channel. Although there is an increase in computational complexity for decoding of non-binary LDPC codes, this could be appropriately addressed with the rapid development of hardware in terminals.

Fig 6

Performance comparison of LDPC codes over GF(2) and GF(64) on AWAN channel.

Robustness analysis

Moreover, the robustness of the proposed initialization by considering LDPC codes with different lengths is analyzed. Fig 7 shows the performance comparison of the proposed approach with QC LDPC codes (88, 44) and (200, 100) over GF(64) on AWAN channel with A = 0.1 and Γ = 0.1. We can observe that, for both long and short codes, the decoding algorithm with the optimized initialization can be performed accurately and QC LDPC codes (200, 100) has 1 dB coding gain over QC LDPC codes (88, 44) at BER = 10−5. It can be concluded that, the proposed initialization shows an excellent robustness.

Fig 7

Performance comparison of QC LDPC codes on AWAN channel with A = 0.1 and Γ = 0.1.

Effect of channel parameters

We also investigate the effect of different AWAN channel parameters on performance of the proposed FFT BP decoding algorithm. Table 1 summarizes the signal-to-noise ratio E/N0 corresponding to BER = 10−5 under the variation of the impulsive coefficient A and the GIR Γ respectively. From this table, it is clear that E/N0 becomes larger as the impulsive coefficient A increases. The reason is that, the larger A is, the more continuous impulsive noises would be, i.e., the statistical characteristics of the Class A noise approximate these of the Gaussian noise. Thus, the impulsive noise would not be suppressed well which leads to a worse performance of the optimized decoding algorithm. Particularly, the proposed algorithm with A = 10 suffers from 10.5 dB degradation compared to that with A = 0.01 at BER = 10−5.

Additionally, Table 2 also shows that E/N0 becomes larger as the GIR Γ increases from 0.01 to 1. It is because that, with a larger Γ, the Class A noise shows fewer impulse characteristics and gets similar to the Gaussian noise. Therefore, the impulsive noise would not be suppressed well, which results in a worse performance. In contrast, E/N0 becomes smaller as the GIR increases to 10. The performance is improved due to the fact that the impulse noise power becomes much small as the GIR increases further. Accordingly, the optimized decoding algorithm with Γ = 10 performs about 1.4 dB coding gain over that with Γ = 0.1 at BER = 10−5.

Table 2

Effect of AWAN channel parameters on performance of the optimized FFT BP.

A	0.01	0.1	1	10
Eb/N0 (Γ = 0.1)	2.5 dB	4.2 dB	13 dB	12.9 dB
Γ	0.01	0.1	1	10
Eb/N0 (A = 0.1)	3.6 dB	4.2 dB	4.35 dB	2.8 dB

Simplification

Series truncation of the PDF of the Class A noise is introduced to simplify the calculation in section 4. Further, we compare it with another approach mentioned in [43] by piecewise fitting the optimal LLR in (11). Consider the LLR is related to the channel parameters, two piecewise functions were chosen to fit the optimal LLR with Γ = 0.1 and A = 0.1, which could be expressed as , where z denotes the amplitude of the received signal and is the Gaussian noise power of the Class A noise.

From Fig 8, it can be observed that, the BER performance of LLR with series truncation achieves 0.4 dB coding gain over that of LLR with piecewise fitting. Similar to the previous conclusion, the decoding algorithm with the optimized initialization shows superiority over the competing methods. Although the fitting approach makes the calculation easier, unfortunately, it is difficult to derive the relationship between the fitting function and channel parameters. Hence, we have to recalculate the fitting function under the variation of the channel parameters A and Γ, which complicates the decoding procedure.

Fig 8

Performance comparison of the proposed method with piecewise fitting.

Conclusion

In this paper, we investigate LDPC decoding over GF(q) in impulsive noise environments for modern navigation satellite communication. By jointly considering the Class A noise model and the series truncation, we propose an optimized initialization for LDPC decoding over GF(q) on AWAN channel, which can be employed in BP-based iterative decoding algorithm. In addition, convergence, validity, and robustness of the proposed initialization are analyzed and discussed with extensive experiments. Simulation results demonstrate that, the decoding algorithm with the optimized initialization achieves 12.2 dB coding gain at BER = 10−5 compared to conventional methods on the assigned AWAN channel. Moreover, LDPC codes over GF(q) acquire 1.7 dB coding gain over binary LDPC codes at BER = 10−5 on AWAN channel. Robustness and the effect of channel parameters are confirmed by considering LDPC codes with different lengths and AWAN channel with parameters. Furthermore, by comparing the proposed method and the piecewise fitting method, experimental result verifies the feasibility of our method in practical applications. With the continues development of BDS-3, to employ the optimized initialization for LDPC decoding over GF(q) can achieve superior performance significantly in impulsive noise environments. The optimized initialization proposed in this paper can be also extended to decoding process of other 5G or 6G applications. In the future work, we will focus on studying LDPC decoding over GF(q) in a more complex noise environment for B5G and 6G systems.

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13 Jul 2020

PONE-D-20-15556

An optimized initialization for LDPC decoding over GF(q) in impulsive noise environments

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Reviewer #1: This is a good contribution, however, when you mention in the abstract: In this paper, an optimized initialization by calculating posterior probabilities of received symbols is proposed for non-binary

LDPC decoding over additive white Class A noise (AWAN) channel... the optimized initialization can be performed by applying different optimization technqiues? do you think that metaheuristics are a good option? can you detail this in the abstract?

In the second paragraph: ...For current navigation system, LDPC codes over GF(q) is adopted in encoding

and decoding. With the development of BeiDou Navigation Satellite System (BDS) 3,

B1C signal and B2a signal have been utilized gradually. The B-CNAV1 navigation

message is transmitted through B1C signal. Each frame of message consists of 3 sub

frames, of which the second sub frame exploits LDPC codes (200,100) over GF(64)

and the third sub frame leverages LDPC codes (88, 44) over GF(64) for encoding [7].

Meanwhile, the B-CNAV2 navigation message is transmitted through B2a signal with

LDPC codes (96, 48) over GF(64). Nowadays, evaluation and analysis of LDPC codes

over GF(q) for navigation satellite communication, especially in harsh environments,

arouse attentions of researchers all over the world....

When you conclude this paragraph, can you detail why in harsh environments,

arouse attentions of researchers all over the world? are there especial circumstances to consider these LDPC and GF the state of the art?

In section: Conventional Initialization in Decoding Process.. you conclude: And the initial messages of BP algorithm over GF(q) could be obtained by bring

(8) into (5)...again, can you detail why BP is better thab GF?, in addition, it seems that GF is the reference to be improved,

In section: An optimized initialization approach for bp decoding

over gf(q) on awan channel, you describe step by step and include simulation results, but can you include a pseudocode? see examples of pseudocode in this paper : Optimizing the kaplan–yorke dimension of chaotic oscillators applying de and pso, 2019

Can you discuss convergence issues on your optimization initialization? see for example a cese of study in this paper: Convergence rates of the efficient global optimization algorithm for improving the design of analog circuits, 2020

Reviewer #2: 1. Results and findings can be improved further. Interesting to improve the conclusion section as well.

2. Add some key references such as: Deep Transfer Learning based Classification Model for COVID-19 Disease, Classification of COVID-19 patients from chest CT images using multi-objective differential evolution--based convolutional neural networks, Efficient Prediction of Drug-drug interaction using Deep Learning Models, Multi-objective particle swarm optimization-based adaptive neuro-fuzzy inference system for benzene monitoring, Improved Particle Swarm Optimization Based Adaptive Neuro-Fuzzy Inference System for Benzene Detection, Fusion of medical images using deep belief networks

3. Add a suitable diagrammatic flow of the proposed model

4. Authors should add atleast 5 references, otherwise it seems out of scope of PLOS one journal.

**********

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Reviewer #1: No

Reviewer #2: No

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7 Aug 2020

We would like to thank the Editor and the reviewers for their valuable comments and feedback that has helped us make improvements to this paper. We give the respond as an attachment "Response_to_Reviewers.doc".

Reviewer#1, Concern # 1: This is a good contribution, however, when you mention in the abstract: In this paper, an optimized initialization by calculating posterior probabilities of received symbols is proposed for non-binary LDPC decoding over additive white Class A noise (AWAN) channel... the optimized initialization can be performed by applying different optimization techniques? Do you think that metaheuristics are a good option? Can you detail this in the abstract?

Author response: Thanks for this comment. In our manuscript, we analyze problems of the existing decoding algorithms and focus on an optimized initialization by calculating posterior probabilities of received symbols or non-binary LDPC decoding on AWAN channel and. The authors agree that we should consider and discuss the metaheuristic as an alternative to make the article more complete. Metaheuristics are widely recognized as efficient approaches for optimization problems. However, as illustrated in [33], the successful application of metaheuristics requires to find a good initial parameter setting, which is a tedious and time consuming task. Moreover, the performance of metaheuristics deteriorates quickly as the dimensionality increases, nevertheless high-dimensional circumstances are extremely common in encoding and decoding. Therefore, in contrast to metaheuristics, the proposed optimized initialization shows superiority in dealing with decoding problems based on BP.

Author action: We have reorganized the manuscript to make it more clearly. We updated by rewriting the manuscript and adding a new section (i.e. Section 2 “Related work”). And we have included the consideration of metaheuristics in details in Section 2 as follows:

This paper is focused on the design of an optimized initialization for non-binary LDPC codes on AWAN channel. In contrast to metaheuristics, our optimized initialization is dedicated to deal with decoding problems based on BP. Generally, as the famous optimization techniques, metaheuristics are widely recognized as efficient approaches for optimization problems, such as particle swarm optimization (PSO) [26, 27] and differential evolution (DE) [28]. Various metaheuristic methods have reported advantages in image segmentation [29], tuning hyper-parameters of deep neural networks [30, 31], and benzene prediction model [32]. However, as illustrated in [33], the successful application of metaheuristics requires to find a good initial parameter setting, which is a tedious and time consuming task. Moreover, the performance of metaheuristics deteriorates quickly as the dimensionality increases, nevertheless high-dimensional circumstances are extremely common in encoding and decoding. Therefore, decoding algorithms based on BP are considered for non-binary LDPC codes on AWAN channel, and an optimized initialization by calculating posterior probabilities of received symbols is proposed.

________________________________________

Reviewer#1, Concern # 2: In the second paragraph: “...For current navigation system, LDPC codes over GF(q) is adopted in encoding and decoding. With the development of BeiDou Navigation Satellite System (BDS) 3, B1C signal and B2a signal have been utilized gradually. The B-CNAV1 navigation message is transmitted through B1C signal. Each frame of message consists of 3 sub frames, of which the second sub frame exploits LDPC codes (200,100) over GF(64) and the third sub frame leverages LDPC codes (88, 44) over GF(64) for encoding [7]. Meanwhile, the B-CNAV2 navigation message is transmitted through B2a signal with LDPC codes (96, 48) over GF(64). Nowadays, evaluation and analysis of LDPC codes over GF(q) for navigation satellite communication, especially in harsh environments, arouse attentions of researchers all over the world....”. When you conclude this paragraph, can you detail why in harsh environments,

arouse attentions of researchers all over the world? Are there especial circumstances to consider these LDPC and GF the state of the art?

Author response: Thanks for this comment. The authors agree that we should illustrate the especial circumstances to consider LDPC and GF the state of the art and describe why navigation communication arouses attentions of researchers in harsh environments.

Author action: We have rewritten the Section 1 in our manuscript and added more descriptions about the above issues in the second and third paragraphs as follows:

Non-binary LDPC codes have been recognized as a powerful technology to encode navigation information efficiently. With the continuous development of BDS 3 and the increasing demand for navigation, people hope to access precise navigation information in time. However, working in severe environments, such as underwater environment surrounded by acoustical noises or in the power industry, the decoder of non-binary LDPC codes is inevitably subject to impulsive interference, which causes poor reliability of transmitted information and even leads to communication failure. Therefore, nowadays, evaluation and analysis of LDPC codes over GF(q) for navigation satellite communication, especially in harsh environments, arouse attentions of researchers all over the world.

Conventionally, the LDPC decoding algorithms are designed for AWGN channel. However, navigation satellite communication often suffers from irregular noises due to the occurrence of high amplitude “spikes”. For navigation satellite communication, such spikes can be generated in atmosphere where lightning discharges in the vicinity of the receiver, or underwater environment where the ambient acoustical noises includes impulses due to noisy aquatic animals such as snapping shrimp [8-11]. At this time, the original assumption that noises are the Gaussian noises is inadequate. Unfortunately, few attention has been given to decoding algorithms for LDPC codes over GF(q) in impulsive noise environments.

________________________________________

Reviewer#1, Concern # 3: In section: Conventional Initialization in Decoding Process. You conclude: And the initial messages of BP algorithm over GF(q) could be obtained by bring (8) into (5) ...again, can you detail why BP is better than GF? In addition, it seems that GF is the reference to be improved,

Author response: Thanks for this comment. As illustrated in our manuscript, a significant amount of research has been concentrated on the design, encoding, decoding and performance analysis of LDPC codes over GF(q). And iterative decoding algorithm based on BP is an important soft decision decoding algorithm. In section “Conventional Initialization in Decoding Process”, we give the conventional initialization of BP decoding algorithm considering non-binary LDPC codes. Non-binary LDPC codes mean LDPC codes over GF(q). By adding (8) to (5), the initial messages of BP algorithm for non-binary LDPC codes could be obtained.

Author action: We have enhanced the detailed description about why non-binary LDPC codes perform better compared to binary LDPC codes in Section 1 (the penultimate paragraph) and Section 3 as follows:

Generally, compared with binary LDPC, LDPC codes over GF(q) show superior performance in resisting burst errors such as impulsive noises for the characteristic of inner interleaving [12, 13], which makes non-binary LDPC codes more suitable for navigation communication. Furthermore, non-binary LDPC codes combined with q-ary modulation can increase transmission rate obviously [14, 15]. Although there is an increment of computational complexity by adopting LDPC codes over GF(q), with the development of terminal computation, non-binary LDPC codes have become a hot research topic and would become more prevalent and applicable. Therefore, we focus on non-binary LDPC codes decoding on additive white Class A noise (AWAN) channel and present an optimized initialization for decoding in this paper. With simulation experiments, we demonstrate the efficiency of the proposed algorithm.

Non-binary LDPC codes can be considered as a kind of linear block codes which are the extension of binary LDPC codes over GF(q). The difference between non-binary LDPC codes and binary LDPC codes is that each non-zero element of sparse parity-check matrix needs to be obtained from GF(q). The Tanner graph of non-binary LDPC codes given by a sparse parity-check matrix over GF(q) is constructed in the same way as that of binary LDPC codes. Compared with binary LDPC codes, non-binary LDPC codes perform better in communication due to advantages in resisting burst error and high transmission rate [34-37].

The GF(q) related references include [12-15, 34-37].

Also, we have enhanced the detailed description about BP decoding and initialization in Section 3 as:

The conventional decoding methods degrade severely while facing the impulsive noise since those methods acquire the posterior probability of received symbols by making use of the transition probability on AWGN channel. To tackle this problem, based on the BP decoding algorithm, we present an optimized initialization for LDPC decoding over GF(q) on AWAN channel with series truncation.

As mentioned above, the initialization is the crucial part of the BP decoding over GF(q) algorithm. The posterior probability of received symbols is considered as the initial message transmitted from the variable nodes to the check nodes. Thus, to evaluate the posterior probability of received symbols correctly is vital for decoding.

During the initialization of conventional decoding process, we obtain the posterior probability of the received bits from the transition probability of the AWGN channel as equation (9), which suffers from degradation of the bit error rate (BER) in impulsive noise environments. Therefore, for LDPC decoding over GF(q) on AWAN channel, since the channel parameters are only leveraged in the decoding initialization, we need to improve the initialization of the iterative decoding process without changing the information transmission mode between variable nodes and check nodes. Under this circumstance, the optimized initialization can be applied to various traditional LDPC decoding algorithms over GF(q), such as FFT BP and EMS algorithm. Generally, the BP decoding algorithm with optimized initialization for LDPC codes over GF(q) on AWAN channel can be illustrated by the flow chart in Fig 1.

In addition, Figure 1 has been redrawn as follows and new references related to BP (such as Carrasco R A, Johnston M. Non-binary error control coding for wireless communication and data storage. John Wiley & Sons) have been added:

________________________________________

Reviewer#1, Concern # 4: In section: An optimized initialization approach for BP decoding

over GF(q) on AWAN channel, you describe step by step and include simulation results, but can you include a pseudocode? See examples of pseudocode in this paper: Optimizing the kaplan–yorke dimension of chaotic oscillators applying de and pso, 2019

Author response: Thanks for this comment very much. The authors agree that the pseudocode of the proposed initialization is significantly important.

Author action: The authors have read the paper “Optimizing the kaplan–yorke dimension of chaotic oscillators applying de and pso” carefully. The pseudocode in this paper seems very clear and powerful. Moreover, this paper introduces the optimization of the Kaplan-Yorke dimension of chaotic oscillators by applying metaheuristics such as DE and PSO algorithms, which is consistent with our discussion about metaheuristics in Section 2.

We updated the manuscript by adding this paper as reference. And we have included the symbol description and pseudocode of FFT BP decoding with the proposed initialization for non-binary LDPC codes on AWAN channel instead of the original descriptions step by step in Section 4 as:

FFT BP decoding with the proposed initialization (i.e. the optimized FFT BP algorithm) for non-binary LDPC codes on AWAN channel is illustrated in Algorithm 1, …..(in the 5-th paragraph in Section 4)

And the correspond pseudocode is added as:

________________________________________

Reviewer#1, Concern # 5: Can you discuss convergence issues on your optimization initialization? See for example a case of study in this paper: Convergence rates of the efficient global optimization algorithm for improving the design of analog circuits, 2020.

Author response: Thanks for this comment very much. The authors agree that there should be more discussions with respect to the convergence of the proposed algorithm. We have conducted researches on the recommended article “Convergence rates of the efficient global optimization algorithm for improving the design of analog circuits” and found that convergence rates and comparisons performed by this article were useful.

Author action: We have conducted researches on the recommended articles and performed experiments with respect to the convergence. We have reorganized the manuscript and added a new subsection “Convergence comparison” in Section 5 “Results and discussion” as follows:

Firstly, convergence issues have been considered as in [40] and performance of different initializations was evaluated utilizing the BER. To be specific, we compared the BER of the FFT BP decoding algorithm with the optimized initialization (i.e. the optimized FFT BP algorithm) with that of the conventional FFT BP algorithm (designed for AWGN channel) on AWAN channel. We should note that the difference between the two algorithms is the initialization. Therefore, the statistical evaluation was performed using the Wilcoxon rank test metric runs of both algorithms for QC LDPC codes (88, 44) with the statistical significance value. The null hypothesis H0 is ‘The difference between obtained by the optimized FFT BP algorithm and the conventional FFT BP algorithm is identical with the same BER’. Meanwhile, the alternative hypothesis is set as ‘the optimized FFT BP algorithm is validated’.

Table 1 presents the convergence comparisons of different initializations using the Wilcoxon Signed-Rank Test metric at different BER, where the ‘+’ indicates the cases when the algorithm acquires better coding gain. It clearly shows that the FFT BP algorithm with the proposed optimized initialization is statistically more superior.

And results of convergence comparisons at different BER by utilizing Wilcoxon rank test is also added as Table 1.

________________________________________

Reviewer#2, Concern # 1: Results and findings can be improved further. Interesting to improve the conclusion section as well.

Author response: Thanks for this comment very much. The authors agree that both section should be improved further. We have rewritten Section 5 “Results and discussion” and Section 6 “Conclusion”

Author action: We have updated by rewriting all experimental results to make them clear and adding a new subsection (i.e. convergence comparison) in Section 5, which has been described in Reviewer#1, Concern # 5.

Further, we have rewritten Section 6 “Conclusion” as:

In this paper, we investigate LDPC decoding over GF(q) in impulsive noise environments for modern navigation satellite communication. By jointly considering the Class A noise model and the series truncation, we propose an optimized initialization for LDPC decoding over GF(q) on AWAN channel, which can be employed in BP-based iterative decoding algorithm. In addition, convergence, validity, robustness of the proposed initialization are analyzed and discussed with extensive experiments. Simulation results demonstrate that, the decoding algorithm with the optimized initialization achieves 12.2 dB coding gain at BER=10-5 compared to conventional methods on the assigned AWAN channel. Moreover, LDPC codes over GF(q) acquire 1.7 dB coding gain over binary LDPC codes at BER = 10-5 on AWAN channel. Robustness and the effect of channel parameters are confirmed by considering LDPC codes with different lengths and AWAN channel with parameters. Furthermore, by comparing the proposed method and the piecewise fitting method, experimental result verifies the feasibility of our method in practical applications. With the continues development of BDS-3, to employ the optimized initialization for LDPC decoding over GF(q) can achieve superior performance significantly in impulsive noise environments. In the future work, we will focus on studying LDPC decoding over GF(q) in a more complex noise environment.

________________________________________

Reviewer#2, Concern # 2: Add some key references such as: Deep Transfer Learning based Classification Model for COVID-19 Disease, Classification of COVID-19 patients from chest CT images using multi-objective differential evolution--based convolutional neural networks, Efficient Prediction of Drug-drug interaction using Deep Learning Models, Multi-objective particle swarm optimization-based adaptive neuro-fuzzy inference system for benzene monitoring, Improved Particle Swarm Optimization Based Adaptive Neuro-Fuzzy Inference System for Benzene Detection, Fusion of medical images using deep belief networks

Author response: Thanks for this comment very much. The authors agree that there should be more related references in this manuscript. Since our manuscript introduces an optimized initialization, as the famous optimization techniques, the authors agree that the metaheuristic should be considered and discussed as an alternative to make the article more complete. We have conducted researches on the recommended articles and found that those excellent articles are closely related to our consideration of metaheuristic algorithms. “Classification of COVID-19 patients from chest CT images using multi-objective differential evolution--based convolutional neural networks” presents a CNN to classify the chest CT images as as infected (+ve) or not (-ve) and tunes the initial parameters of CNN with multi-objective differential evolution. “Deep Transfer Learning based Classification Model for COVID-19 Disease” and “Fusion of medical images using deep belief networks” combine deep learning with meta-heuristics and give a detailed description about tuning hyper-parameters of deep neural networks. “Improved Particle Swarm Optimization Based Adaptive Neuro-Fuzzy Inference System for Benzene Detection” extends the metaheuristic algorithm to benzene prediction model. “Multi-objective particle swarm optimization-based adaptive neuro-fuzzy inference system for benzene monitoring” employs the metaheuristic (PSO) to enhance the accuracy of ANFIS for runtime parameter.

Author action: We have conducted researches on the recommended articles and included those excellent literature as our references. We updated by rewriting the manuscript and adding a new section (i.e. Section 2 “Related work”). We have included the consideration of metaheuristics in details with the recommended articles as important references in Section 2 as:

Generally, receivers adopts parameters of the AWGN channel and conventional decoding methods for LDPC decoding, which results in serious degradation in impulsive noise environments. The impulsive component of interference has been found to be significant which influences the reliability of transmitted information.

Various attempts have been made to develop models of impulsive noises that can be divided into empirical models and physical models. Class A noise model proposed by Middleton is a typical kind of physical models [23]. The statistical feature of Class A noise is much different from that of Gaussian noise, therefore the LDPC decoding algorithms for AWGN channel are not suitable for Class A noise environments. Maad et al. analyzed the performance of LDPC codes in heavy-tailed, symmetric alpha stable noise (SαS) channels [24]. Further, Nakagawa et al. proposed the sum-product decoding method for binary LDPC codes in Class A noise environment [25]. However, few researches have ever explored decoding algorithms for LDPC codes over GF(q) in Class A noise environments.

This paper is focused on the design of an optimized initialization for non-binary LDPC codes on AWAN channel. In contrast to metaheuristics, our optimized initialization is dedicated to deal with decoding problems based on BP. Generally, as the famous optimization techniques, metaheuristics are widely recognized as efficient approaches for optimization problems, such as particle swarm optimization (PSO) [26, 27] and differential evolution (DE) [28]. Various metaheuristic methods have reported advantages in image segmentation [29], tuning hyper-parameters of deep neural networks [30, 31], and benzene prediction model [32]. However, as illustrated in [33], the successful application of metaheuristics requires to find a good initial parameter setting, which is a tedious and time consuming task. Moreover, the performance of metaheuristics deteriorates quickly as the dimensionality increases, nevertheless high-dimensional circumstances are extremely common in encoding and decoding. Therefore, decoding algorithms based on BP are considered for non-binary LDPC codes on AWAN channel, and an optimized initialization by calculating posterior probabilities of received symbols is proposed.

________________________________________

Reviewer#2, Concern # 3: Add a suitable diagrammatic flow of the proposed model

Author response: Thanks for this comment. The authors agree that the pseudocode of the proposed initialization is significantly important.

Author action: We redrawn Fig. 1 (the flow chart of BP decoding for LDPC codes over GF(q) on AWAN channel) and added the pseudocode of the proposed model in our manuscript, which has been described in Reviewer#1, Concern # 4.

Also, to make the manuscript more clear and complete, we have enhanced Section 1 with a detailed description of the contribution as follows:

Therefore, we focus on non-binary LDPC codes decoding on additive white Class A noise (AWAN) channel and present an optimized initialization for decoding in this paper. With simulation experiments, we demonstrate the efficiency of the proposed algorithm. The main contributions are summarized as follows:

1) We investigate the problem of LDPC decoding in impulsive noise environments for navigation communication and formalize the impulsive noise as the Class A noise model.

2) We propose an optimized initialization by calculating posterior probabilities of received symbols for non-binary LDPC decoding on AWAN channel, which makes use of series truncation for computing effectively.

3) Extensive experiments are conducted on convergence, validity, robustness. The experimental results reveal that the optimized initialization has a significant effect on the decoding performance for non-binary LDPC codes on AWAN channel.

________________________________________

Reviewer#2, Concern # 4: Authors should add at least 5 references, otherwise it seems out of scope of PLOS one journal.

Author response: Thanks for this comment. The authors agree that there should be more related references in this manuscript. We have conducted researches on the recommended articles.

Author action: We have conducted researches on the recommended articles and included those excellent literature as our references, which has been described in Reviewer#2, Concern # 2.

________________________________________

Submitted filename: Response_to_Reviewers.doc

Click here for additional data file.

8 Apr 2021

PONE-D-20-15556R1

An optimized initialization for LDPC decoding over GF(q) in impulsive noise environments

PLOS ONE

Dear Dr. Zhao,

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Nanchang University, China

Academic Editor

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Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice.

Additional Editor Comments (if provided):

The paper seems to be revised well. All the comments are addressed. There is one minor revisions to be noticed. LDPC decoding is a very old technique, so please add some new backgroud and references for that, like 5G or 6G application related references. For example,

[1] “Massively Distributed Antenna Systems With Nonideal Optical Fiber Fronthauls: A Promising Technology for 6G Wireless Communication Systems,” IEEE Vehicular Technology Magazine, Dec. 2020.

[2] "What should 6G be?" Nature Electronics, Jan. 2020.

and so.

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Comments to the Author

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Reviewer #1: All comments have been addressed

**********

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The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

**********

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Reviewer #1: Yes

**********

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Reviewer #1: Yes

**********

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Reviewer #1: the revised paper can be accepted now, the authors have addressed all recommendations of this reviewer

**********

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14 Apr 2021

Thanks editors. The experimental data involved in this manuscript have been uploaded as the Supporting Information files regarding to each experiment (saved in MATLAB data format, i.e. "*.mat").

Additional Editor Comments (if provided): The paper seems to be revised well. All the comments are addressed. There is one minor revisions to be noticed. LDPC decoding is a very old technique, so please add some new background and references for that, like 5G or 6G application related references. For example, [1] "Massively Distributed Antenna Systems with Nonideal Optical Fiber Fronthauls: A Promising Technology for 6G Wireless Communication Systems," IEEE Vehicular Technology Magazine, Dec. 2020, [2] "What should 6G be?" Nature Electronics, Jan. 2020; and so.

Author response: Thanks for this comment very much. The authors agree that there should be more background and references related to 5G or 6G application in our manuscript. As LDPC decoding is an old technique, we hope to highlight the connection between this paper and the latest communication system.

Author action: We have made the manuscript more clearly. We updated by including some new background in Section 1 “Introduction” and adding some related references to emphasize the 5G or 6G application as follows:

With the explosive development of communication technology, the new mobile communication systems, such as beyond fifth generation (B5G) and sixth generation (6G) systems, will suffer from severe challenges imposed by the requirement for heavy connection density and high efficiency [1]. Especially for 6G, satellite communications play an important role in providing high quality communication services to achieve the worldwide connectivity [2]. As one of the critical components, …

[1]. Yu L, Wu J, Zhou A, et al. Massively Distributed Antenna Systems With Nonideal Optical Fiber Fronthauls: A Promising Technology for 6G Wireless Communication Systems. IEEE Vehicular Technology Magazine. 2020; 15(4): 43-51.

[2]. Dang S, Amin O, Shihada B, et al. What should 6G be?. Nature Electronics. 2020; 3(1): 20-29.

Further, in Section 6 “Conclusion”, we have included the consideration of the next communication systems as follows:

The optimized initialization proposed in this paper can be also extended to decoding process of other 5G or 6G applications. In the future work, we will focus on studying LDPC decoding over GF(q) in a more complex noise environment for B5G and 6G systems

Submitted filename: Response_to_Reviewers.doc

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19 Apr 2021

An optimized initialization for LDPC decoding over GF(q) in impulsive noise environments

PONE-D-20-15556R2

Dear Dr. Zhao,

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22 Apr 2021

PONE-D-20-15556R2

An optimized initialization for LDPC decoding over GF(q) in impulsive noise environments

Dear Dr. Zhao:

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on behalf of

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