Improved dragonfly algorithm and residual degeneration combined LDPC (Low Density Parity Check) quantitative hierarchical decoding working method

Abstract

The invention relates to an LDPC (Low Density Parity Check) quantitative hierarchical decoding working method combining an improved dragonfly algorithm with residual degradation, and belongs to the technical field of wireless communication. The method specifically comprises the steps that firstly, a pilot signal is sent by a signal source, average least square estimation of channel state information is obtained through analysis based on representation of a received signal in a time domain and a frequency domain, noise power is determined, then the pilot signal is introduced into an improved dragonfly algorithm, quantization parameter set training is carried out, and the improved dragonfly algorithm reads a sample signal to obtain the channel state information. And finally, applying the self-adaptive quantization parameter set obtained by training to a residual degradation decoding algorithm, and if a decoding result does not meet a verification equation, carrying out next iteration. The performance of the residual degradation decoding algorithm is superior to that of the DLNMS, hardware resources are saved, the decoding performance of layered quantization is improved, the decoding time is shortened, and therefore the decoding speed is increased.

Description

Technical Field

[0001] The invention relates to an improved dragonfly algorithm combined with an LDPC quantization layered decoding working method for residual degradation, and belongs to the technical field of wireless communication. Background Art

[0002] As people's demand for mobile communications has increased significantly, mobile communication technology has developed rapidly. The currently widely used fifth-generation mobile communication systems (5G) have put forward higher requirements for data transmission rate, reliability and low latency. Low-density parity-check code (LDPC) has been adopted by the 3GPP standard as the data channel coding scheme for 5G enhanced mobile broadband (eMBB) scenarios due to its excellent error correction performance and parallel decoding characteristics.

[0003] LDPC codes were proposed by Gallager in 1962. Their decoding algorithms are mainly divided into hard decision and soft decision decoding algorithms (see Gallager R. Low-density parity-check codes[J]. IRE Transactions on Information Theory, 1962, 8(1): 21-28.). Hard decision algorithms usually use bit flipping algorithms, which are easy to implement in hardware but have poor performance. Soft decision algorithms usually use belief propagation (BP) algorithms, which have good error correction performance but high implementation complexity. Based on this, Fossorier et al. improved the BP algorithm and proposed the minimum sum algorithm (Min-Sum, MS), which greatly reduced the algorithm complexity with a small performance loss (see MPC Fossorier, M. Mihaljevic, H. Imai. Reduced complexity iterative decoding of low-density parity check codes based on belief propagation. IEEE Transactions on Communications, 1999, 47(5): 673-680.). Subsequently, Chen et al. studied the MS algorithm and proposed the Normalized Min-Sum (NMS) algorithm and the Offset Min-Sum (OMS) algorithm with the help of density evolution theory (see Chen J., Dholakia A., Eleftheriou E., et al. Reduced-complexity decoding of LDPC codes using normalized and offset min-sum algorithms[J]. IEEE Transactions on Communications, 2005, 53(8): 1288-1299.), which further approached the performance of the BP algorithm.In 2004, Hocevar proposed the layered BP decoding algorithm to reduce decoding latency and storage requirements (see Hocevar D E. A reduced complexity decoder architecture via layered decoding of LDPC codes[C] / / IEEE Workshop on Signal Processing Systems. 2004: 107-112.). Subsequently, the layered normalized min-sum (LNMS) algorithm was applied. To further conserve hardware resources, C. CHEN and SY. Lin proposed the Delete Layer-NMS (DLNMS) algorithm based on the layered algorithm, but at the expense of some decoding performance (see Chen C., Lin SY. A trimmed belief-propagation decoding for LDPC codes[C] / / IEEE International Symposium on Information Theory. 2008: 487-491.).

[0004] Similarly, to improve the speed of the decoding algorithm, a quantization strategy is usually adopted for the BP algorithm or the NMS algorithm. However, when the traditional uniform quantization strategy is combined with the LNMS algorithm, overflow usually occurs, resulting in serious decoding errors (see Layered Decoding Algorithm Based on Improved Integer Quantization LDPC Code [J]. Northwestern Polytechnical University, 2014, 32(6): 974-980.). To solve the performance loss after quantization, the existing technology proposes a quantization minimum and LDPC decoding research based on deep learning, which improves the performance of quantization decoding (see Research on Quantization Minimum and LDPC Decoding Based on Deep Learning [D]. Shanghai Jiao Tong University, 2020. DOI:10.27307 / d.cnki.gsjtu.2020.002450.). However, it also brings a large amount of pre-training and network design complexity, and has higher requirements on hardware. At the same time, the quantization method is not applied to the layered decoding algorithm. Summary of the Invention

[0005] In view of the shortcomings of the existing technology, the present invention provides an improved dragonfly algorithm (IDA) combined with residual degradation LDPC quantization layered decoding working method, using the residual degradation decoding algorithm (RB-DLNMS) with better decoding performance than DLNMS, combined with the IDA intelligent optimization algorithm to obtain an adaptive quantization method Q2RB-DLNMS suitable for layered decoding. Integer quantization improves the decoding performance of layered quantization with minimal loss in decoding performance, shortens the decoding time, and thus increases the decoding speed. At the same time, the IDA intelligent optimization algorithm has lower design complexity and less pre-computation than deep learning networks, and has lower hardware requirements, saving a certain amount of hardware resources.

[0006] The technical solutions of the present invention are as follows: An improved dragonfly algorithm combined with residual degenerate LDPC quantization layered decoding method, such as Figure 1 The specific steps are as follows: (1) Sending a pilot signal; (2) Estimating channel state information (CSI); (3) Determine the noise situation; (4) The central computing unit performs intelligent optimization training: the pilot signal is introduced into the improved dragonfly algorithm to train the quantization parameter sets of the two quantization methods. The improved dragonfly algorithm relies on reading a large number of sample signals to automatically find the noise distribution and the relationship between the signal and the noise, thereby dynamically adjusting the quantization parameters to obtain the optimal quantization parameter set (for example, when the signal-to-noise ratio is small, the quantization value density at both ends is larger, and the density of the quantization interval near the zero end is larger to adapt to the signal inversion caused by the larger noise and the low sensitivity of the quantization value); (5) Decoding: Apply the optimal quantization parameter set obtained through training to the residual degradation decoding algorithm.

[0007] According to the preferred embodiment of the present invention, in step (1), specifically, the signal source sends a D-frame signal, assuming that a single frame sends a signal , M is the number of rows of the check matrix during encoding, that is, the number of valid bits, N is the number of columns of the check matrix during encoding, and NM is the number of check bits.

[0008] According to the preferred embodiment of the present invention, in step (2), specifically, the time domain representation of the received signal r is r=h*s+w, and the frequency domain representation is R=HS+W, wherein H is the fast Fourier transform (FFT) of the channel system response function h, S is the fast Fourier transform (FFT) of the transmitted signal s, and W is the fast Fourier transform of the noise w. is the fast Fourier transform of the received signal r, and the average least squares estimate of the channel state information is obtained by analysis: , where D is the number of frames of the received signal and i represents the frame number of the data.

[0009] According to the preferred embodiment of the present invention, in step (3), specifically, the transmitted signal is reconstructed using the channel state information estimation to obtain , and then get the residual , from the frequency domain uniform distribution characteristics of the noise signal, we can know that the noise power It can be estimated by the average power of the residual: , where D is the number of frames of the received signal, i represents the frame number of data, and j represents the bit number of data.

[0010] According to the preferred embodiment of the present invention, in step (4), specifically, the two quantization methods are Q1 (quantization method 1) and Q2 (quantization method 2): Q1: Fix the range of each quantization interval and the number of quantization bits B, change the interval quantization value, limit the quantization value, and quantize the input signal through the parameters of the quantization parameter set to obtain the corresponding quantization value. The quantization diagram can be seen in the attached Figure 2 ; Q2: Fixed the number of quantization bits B, changed the quantization interval range, and quantized the input signal through the parameters of the quantization parameter set to obtain the corresponding quantization value. The quantization diagram can be seen in the attached Figure 3 ; The optimization process of improving the dragonfly algorithm is attached. Figure 4 , the steps are as follows: (41) Optimize parameter initialization assignment, including the maximum number of decoding iterations , optimize the number of iterations , population size , Euclidean distance threshold , differential evolution rate , and perform tent chaos initialization on the two quantization parameter sets. The steps are as follows: Normalize first: (1) Then perform tent mapping: (2) Finally, denormalize: (3) In formula (1)(2)(3), Indicates type The parameter vector of and Represents the upper and lower limit vectors of the parameter vector, represents the normalized parameter vector. Represents a random number between 0 and 1. Indicates the length of the parameter vector of this type, express The parameter vector after Tent mapping, express The parameter vector after denormalization; (42) The leading signal is transformed into the logarithmic domain, ,in, Indicates the received Frame No. bits of information value, Indicates that the received Frame No. The logarithmic domain information value after the information value of the bits is converted into the logarithmic domain; Then perform quantitative processing of the two strategies to obtain quantitative data ; (43) Importing the quantized data into the residual degradation decoding algorithm for decoding, see the specific implementation of the residual degradation decoding algorithm below for details; (44) Calculate bit error rate and frame error rate; (4) (5) in, is the number of error-decoded bits, is the total number of decoded bits, is the number of error-decoded frames, is the total number of decoded frames; (45) Adjust the food source and natural enemies according to the bit error rate and frame error rate, perform differential evolution, adjust the parameters, calculate the displacement vector, and perform the next adjustment iteration of the quantization parameters. The specific steps are as follows: When the number of iterations When , the parameter vector population behavior is updated, and the process is as follows: Separate expressions: (6) in, for The position vector to avoid collision with other parameter vectors, for The total number of parameter vectors nearby, for Nearby parameter vector, and the distance is determined by Euclidean distance , The parameter vector Wei, and when When lie in nearby; Alignment expressions: (7) in, for parameter vector for the alignment behavior, for nearby The difference parameter vector of For the The iteration vector; Aggregate expressions: (8) in, for parameter vector of the aggregation behavior; Foraging and avoiding predators expressions: (9) in, for The parameter vector of the foraging behavior, for The parameter vector of the behavior of avoiding natural enemies makes the decoded The minimum quantization parameter vector is , accordingly The maximum quantization parameter vector is , if exists Equal multiple parameter vectors The smallest act , accordingly The greatest achievement ; Parameter vector update expression: (10) in, is the separation weight; is the alignment weight; is the aggregation weight; is the food weight; For natural enemy weight; is the inertia weight, and the above specific values ​​are all empirical values; like There are no other parameter vectors nearby, that is, hour, ,in, The function is: (11) in, is the gamma function, is the diffusion factor, and for , is the control factor, generally taken as 1.5; When the number of iterations , the following differential evolution operation is performed: Mutation process expression: (12) in, For the After the iterative mutation parameter vector, is a parameter vector randomly selected from all parameter vectors, is the scaling factor; Cross-process expression: (13) in, For the After the iterative crossover parameter vector, is the crossover rate; Select the process expression: (14) (46) If the iteration is completed, the optimal quantization parameter set is output, otherwise return to step (42).

[0011] According to the present invention, in step (4), the Q1 quantization parameter set: each quantization interval range , interval quantization value , where n is determined by the total length of the quantization interval and the quantization step size, as well as the quantization limit rate , which means the range limit rate of each quantization interval. Through the above two sets of parameters, the quantization value range of each interval under the uniform quantization interval can be obtained as ,in, , is the number of quantization bits, To receive the logarithmic domain information of the data, the quantization processing steps are: by reading the input signal value, determine whether it is in the quantization interval to The signal value after quantization is ; Q2 quantization parameter set: range of each quantization interval , and the number of quantization bits ,in, The total length of the quantization interval and the quantization step are determined; the quantization processing steps are: by reading the input signal value, determining whether it is in the quantization interval to The signal value after quantization is .

[0012] According to the preferred embodiment of the present invention, in step (5), the specific implementation of the residual degradation decoding algorithm involved in training and testing is: (51) Initialization: (15) (16) The initialization process is to receive the logarithmic domain information It is handed over to the 0th layer and is only used in the first iteration. The posterior information is then passed between layers and updated iteratively, where For the check nodes, For the The check node sends The posterior message transmitted by the variable node, For the The variable node to the The posterior message delivered by the verification nodes; (52) Iterative message update: (17) (18) (19) in, For the iterations, This is the first layer, is the correction factor of the minimum sum algorithm, Representatives and All variable nodes connected to the check node, Representatives except This variable node is the same as the All check nodes connected to the check node; (53) Determination of reliable layer: when It is determined to be a reliable layer when , otherwise it is an unreliable layer, where The confidence threshold for reliable layer determination; Determine the confidence threshold based on the set reliability layer Determine whether it belongs to a reliable layer. If it is a reliable layer, skip this layer in the next iteration; (54) Residual degradation: when When , the reliable layer degenerates into an unreliable layer, i.e., an oscillating layer, where is the number of oscillation layers currently existing, is the minimum number of oscillation layers, is the residual threshold; (55) Decoding judgment: when The judgment is , otherwise the judgment is , the decoding results are stored in the matrix middle; (56) Verification results: Substitute the decoding result into the check equation , if the check equation is satisfied, the decoding is output, otherwise it goes to the next iteration.

[0013] The beneficial effects of the present invention are: The residual degradation decoding algorithm proposed in this invention based on inter-layer residual and limited oscillation layer number has better decoding performance than DLNMS and has the advantage of saving certain hardware resources. At the same time, an adaptive quantization method suitable for layered decoding is obtained through the IDA intelligent optimization algorithm. Integer quantization improves the decoding performance of layered quantization with less loss of decoding performance, shortens the decoding time, and thus increases the decoding speed. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 This is a complete system flow chart of the present invention.

[0015] Figure 2 This is a schematic diagram of Q1 quantization in the present invention. The quantization value interval is limited by Control, as shown in the figure When , each quantization value restriction interval just covers all quantization values ​​and does not intersect with each other, and The larger the value, the more the quantization value limit intervals intersect.

[0016] Figure 3 This is a schematic diagram of Q2 quantization of the present invention. The lengths of the quantization intervals are different. The inclined dotted line indicates that the quantization value conforms to the linear function as a whole, and its slope is .

[0017] Figure 4 The present invention improves the optimization process of the dragonfly algorithm.

[0018] Figure 5 The simulation results of the improved dragonfly algorithm combined with the residual degradation decoding algorithm and the comparison with other decoding algorithms are given. Figure 5 (a) is the bit error rate of nine decoding algorithms The relationship diagram with the change of signal-to-noise ratio Eb / N0, Figure 5 (b) is the frame error rate of nine decoding algorithms The relationship diagram with the change of signal-to-noise ratio Eb / N0, Figure 5 (c) is the average correct decoding time per frame for the nine decoding algorithms Figure 2 shows the relationship between the signal-to-noise ratio (Eb / N0) and the nine decoding algorithms: the unquantized layered decoding algorithms DLNMS and RB-DLNMS, the uniformly quantized NMS (QNMS) decoding algorithm, the three layered decoding algorithms for the Q1 quantization method (Q1LNMS, Q1DLNMS, and Q1RB-DLNMS), and the three layered decoding algorithms for the Q2 quantization method (Q2LNMS, Q2DLNMS, and Q2RB-DLNMS). DETAILED DESCRIPTION

[0019] The present invention will be further described below with reference to embodiments and accompanying drawings, but is not limited thereto.

[0020] Example 1: This embodiment utilizes The improved dragonfly algorithm is used to obtain the optimal quantization parameter set and is applied to the residual degradation decoding algorithm for simulation. The test parameters are as follows: Set the size of the check matrix when encoding to , that is, the length of the signal sent by a single frame is , the effective signal length is , the check signal length is , the code rate is , the number of rows per layer is , the modulation mode is BPSK, the channel is Gaussian white noise, the maximum number of frames , quantization bit width ; The specific steps are as follows: (1) Set the simulation signal-to-noise ratio to

[0021] (2) Generate 1000 frames of random binary sequence As a pilot signal, and BPSK modulation is performed to obtain the transmission signal ; (3) Generate simulated additive Gaussian white noise according to the signal-to-noise ratio, ,in, is a set of lengths and Equal random Gaussian numbers with distribution coefficients , ; (4) Reconstruct the channel information based on the received pilot signal to determine the noise variance , used to determine the logarithmic domain message and quantization interval range; (5) Import the 1000-frame pilot signal into the IDA algorithm for iterative optimization, set the variables of the IDA algorithm to the input signal, and set the fitness function to the decoding error rate ( Takes precedence over ), and obtain two sets of adaptive quantization parameter sets; (6) Substitute the obtained Q1 and Q2 quantization parameter sets into the three layered decoding algorithms of LNMS, DLNMS, and RB-DLNMS for quantization decoding test, and compare the decoding results with the unquantized DLNMS, RB-DLNMS and uniform quantization NMS (QNMS) decoding algorithms to obtain (bit error rate), (Frame Error Rate), (Average correct decoding time per frame), (Average percentage of savings layer operations) data, where , ; Analyze the decoding results and find the minimum number of oscillation layers. for , residual threshold for , reliable layer determination threshold for , the correction factor of the minimum sum algorithm is for conditions, a) By the attached Figure 5 In (a) and (b), it can be found that the decoding performance of the three layered decoding algorithms using the Q2 quantization method is generally better than that of the Q1 quantization method, that is, the performance of Q2RB-DLNMS is better than that of Q1RB-DLNMS, and the performance of Q2DLNMS is better than that of Q1DLNMS. Moreover, the performance of Q2RB-DLNMS is better than that of Q2DLNMS, which is close to that of the unquantized DLNMS decoding algorithm. b) By the attached Figure 5 (c) It can be found that the average decoding time per frame of the three quantized layered decoding algorithms is much lower than that of the unquantized DLNMS, RB-DLNMS, and QNMS algorithms. This shows that the decoding speed of the three quantized layered decoding algorithms has been improved. c) The average amount of saved layer operations is shown in Table 1. It can be seen from Table 1 that the loss of DLNMS before and after quantization is large, while the loss of RB-DLNMS algorithm before and after quantization is relatively small compared to DLNMS. That is, when the signal-to-noise ratio is high, the average amount of saved layer operations before and after quantization of RB-DLNMS is almost no loss, while the loss before and after quantization of DLNMS is obvious;

[0022] From the above analysis, it can be concluded that the Q2RB-DLNMS algorithm using the Q2 quantization method exhibits a faster decoding speed than the unquantized layered decoding algorithm, while sacrificing a small amount of layer operation savings, and better decoding performance than the Q1RB-DLNMS algorithm and the Q2DLNMS algorithm. That is, the Q2RB-DLNMS algorithm has little decoding performance loss before and after adaptive quantization, greatly improves the decoding speed, and saves a certain amount of hardware resources.

Claims

1. An improved dragonfly algorithm combined with residual degenerate LDPC quantization layered decoding method, characterized in that: The specific steps are as follows: (1) Sending a pilot signal; (2) Estimating channel state information; (3) Determine the noise situation; (4) The central computing unit performs intelligent optimization training: the pilot signal is introduced into the improved dragonfly algorithm to train the quantization parameter sets of the two quantization methods. The improved dragonfly algorithm relies on reading a large number of sample signals to automatically find the noise distribution and the relationship between the signal and the noise, thereby dynamically adjusting the quantization parameters to obtain the optimal quantization parameter set; (5) Decoding: Apply the optimal quantization parameter set obtained through training to the residual degradation decoding algorithm.

2. The improved dragonfly algorithm combined with residual degenerate LDPC quantization layered decoding method according to claim 1 is characterized in that: In step (1), specifically, the signal source sends a D-frame signal, assuming that a single frame sends a signal , M is the number of rows of the check matrix during encoding, that is, the number of valid bits, N is the number of columns of the check matrix during encoding, and NM is the number of check bits.

3. The improved dragonfly algorithm combined with residual degenerate LDPC quantization layered decoding method according to claim 2, characterized in that: In step (2), specifically, the time domain representation of the received signal r is r=h*s+w, and the frequency domain representation is R=HS+W, where H is the fast Fourier transform of the channel system response function h, S is the fast Fourier transform of the transmitted signal s, and W is the fast Fourier transform of the noise w. is the fast Fourier transform of the received signal r, and the average least squares estimate of the channel state information is obtained by analysis: , where D is the number of frames of the received signal and i represents the frame number of the data.

4. The improved dragonfly algorithm combined with residual degenerate LDPC quantization layered decoding method according to claim 3 is characterized in that: In step (3), specifically, the transmitted signal is reconstructed using the channel state information estimation to obtain , and then get the residual , from the frequency domain uniform distribution characteristics of the noise signal, we can know that the noise power It can be estimated by the mean power of the residuals: , where D is the number of frames of the received signal, i represents the frame number of data, and j represents the bit number of data.

5. The improved dragonfly algorithm combined with residual degenerate LDPC quantization layered decoding method according to claim 4 is characterized in that: In step (4), specifically, the two quantization methods are Q1 and Q2: Q1: Fix the range of each quantization interval and the number of quantization bits B, change the interval quantization value, impose restrictions on the quantization value, and quantize the input signal through the parameters of the quantization parameter set to obtain the corresponding quantization value; Q2: fix the number of quantization bits B, change the quantization interval range, and quantize the input signal using the parameters of the quantization parameter set to obtain the corresponding quantization value; Improve the optimization process of the dragonfly algorithm. The steps are as follows: (41) Optimize parameter initialization assignment, including the maximum number of decoding iterations , optimize the number of iterations , population size , Euclidean distance threshold , differential evolution rate , and perform tent chaos initialization on the two quantization parameter sets. The steps are as follows: Normalize first: (1) Then perform tent mapping: (2) Finally, denormalize: (3) In formula (1)(2)(3), Indicates type The parameter vector of and Represents the upper and lower limit vectors of the parameter vector, represents the parameter vector after normalization, Represents a random number between 0 and 1. Indicates the length of the parameter vector of this type, express The parameter vector after Tent mapping, express The parameter vector after denormalization; (42) The leading signal is transformed into the logarithmic domain, ,in, Indicates the received Frame No. bits of information value, Indicates that the received Frame No. The logarithmic domain information value after the information value of the bits is converted into the logarithmic domain; Then perform quantitative processing of the two strategies to obtain quantitative data ; (43) Importing the quantized data into the residual degradation decoding algorithm for decoding; (44) Calculate bit error rate and frame error rate; (4) (5) in, is the number of error-decoded bits, is the total number of decoded bits, is the number of error-decoded frames, is the total number of decoded frames; (45) Adjust food sources and natural enemies based on bit error rate and frame error rate, perform differential evolution, adjust parameters, calculate displacement vectors, and perform the next iteration of quantization parameter adjustment; (46) If the iteration is completed, the optimal quantization parameter set is output, otherwise return to step (42).

6. The improved dragonfly algorithm combined with residual degenerate LDPC quantization layered decoding method according to claim 5, characterized in that: In step (4), Q1 quantization parameter set: the range of each quantization interval , interval quantization value , where n is determined by the total length of the quantization interval and the quantization step size, as well as the quantization limit rate , which means the range limit rate of each quantization interval. The range of quantization values ​​of each interval under the uniform quantization interval is obtained by the above two sets of parameters. ,in, , is the number of quantization bits, To receive the logarithmic domain information of the data, the quantization processing steps are: by reading the input signal value, determine whether it is in the quantization interval to The signal value after quantization is ; Q2 quantization parameter set: range of each quantization interval , and the number of quantization bits ,in, The total length of the quantization interval and the quantization step are determined; the quantization processing steps are: by reading the input signal value, determining whether it is in the quantization interval to The signal value after quantization is 。 7. The improved dragonfly algorithm combined with residual degenerate LDPC quantization layered decoding method according to claim 6, characterized in that: In step (5), the specific implementation of the residual degradation decoding algorithm involved in training and testing is: (51) Initialization: (15) (16) The initialization process is to receive the logarithmic domain information It is handed over to the 0th layer and is only used in the first iteration. The posterior information is then passed between layers and updated iteratively, where For the check nodes, For the The check node sends The posterior message transmitted by the variable node, For the The variable node to the The posterior message delivered by the verification nodes; (52) Iterative message update: (17) (18) (19) in, For the iterations, This is the first layer, is the correction factor of the minimum sum algorithm, Representatives and All variable nodes connected to the check node, Representatives except This variable node is the same as the All check nodes connected to the check node; (53) Determination of reliable layer: when It is determined to be a reliable layer when , otherwise it is an unreliable layer, where The confidence threshold for reliable layer determination; Determine the confidence threshold based on the set reliability layer Determine whether it belongs to a reliable layer. If it is a reliable layer, skip this layer in the next iteration; (54) Residual degradation: when When , the reliable layer degenerates into an unreliable layer, i.e., an oscillating layer, where is the number of oscillation layers currently existing, is the minimum number of oscillation layers, is the residual threshold; (55) Decoding judgment: when The judgment is , otherwise the judgment is , the decoding results are stored in the matrix middle; (56) Verification results: Substitute the decoding result into the check equation , if the check equation is satisfied, the decoding is output, otherwise it goes to the next iteration.

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