LDPC sparse check matrix reconstruction method under high bit error rate

Abstract

The application belongs to the field of channel coding identification, and discloses a high error rate LDPC sparse check matrix reconstruction method, which comprises the following steps: step A, calculating the probability of once extraction containing check nodes, and further determining the number of random extraction; step B, constructing a square matrix by randomly extracting LDPC code bit, and obtaining a suspected check vector by Gaussian elimination; step C, judging the suspected check vector solved by Gaussian elimination in step B based on the statistical characteristics of the relationship of the suspected check vector and the minimum error decision criterion; step D, judging the check vector reserved in step C on the overall received code word matrix, and improving the proportion of error-free code groups in the received data by adopting the method of "eliminating error code words" or "flipping the lowest unreliable bit" according to the number of received code words; step E, repeating steps B to D until the number of iteration extraction is reached; compared with the existing LDPC code check matrix reconstruction algorithm, the application has stronger fault tolerance and lower complexity.

Description

Technical Field

[0001] This invention belongs to the field of channel coding identification, and specifically relates to a method for reconstructing a sparse parity-check matrix of LDPC under high bit error rate. Background Technology

[0002] LDPC (low density parity check code) codes are block codes defined by a parity check matrix, characterized by long code length, sparse parity check matrix, and performance approaching the Shannon limit. For LDPC code recognition research, the long code length often makes traditional analysis methods (matrix analysis, Walsh-Hadamard transform, etc.) computationally too complex, leading to their failure. Therefore, LDPC code recognition has become a challenging problem in channel coding analysis.

[0003] Most research on LDPC code recognition and parity check matrix reconstruction assumes a sufficient number of received codewords. Bao Xin, for example, reconstructs the sparse parity check matrix by using column elimination, parity check vector determination criteria, and asymptotic transformations to find correct parity check vectors and eliminate erroneous code groups. While this method offers some fault tolerance, it requires a large amount of data under error conditions, and the resulting parity check matrix is ​​non-sparse and unusable for decoding. Subsequent research proposed a parity check matrix sparsification algorithm based on second-order and P-order row elimination transformations. This algorithm is practical for sparse matrices with bidiagonal forms, but for non-bidiagonal sparse matrices, the elimination order is high, leading to a sharp increase in computational complexity. Chen Ze, aiming to improve the performance of LDPC parity check matrix reconstruction, introduced an LDPC feedback iterative decoding method. This method uses reconstructed sparse parity check vectors to correct errors in LDPC decoding, significantly improving the algorithm's performance. However, this algorithm requires repeated Gaussian elimination, sparsification, and iterative decoding, resulting in high computational complexity. Wu Zhaojun proposed a novel LDPC sparse parity-check matrix reconstruction method (hereinafter referred to as the Wu algorithm). Utilizing the sparsity of the parity-check matrix, it randomly selects codewords to reduce the dimensionality of the dual space, ultimately reconstructing the sparse parity-check matrix. However, its recognition performance remains unsatisfactory when the number of codewords is small or the bit error rate is high. For cases with a small number of received codewords, Yu Peidong proposed an LDPC code open set recognition algorithm based on finding low-weight codewords. This method searches for low-weight vectors (i.e., the sparse parity-check vectors to be identified) one by one in the dual space of the received code vector space, thereby reconstructing the sparse parity-check matrix. It uses an exponential distribution to model the number of iterations, provides an iteration stopping criterion and computational complexity analysis, and analyzes the number of received code vectors. However, this method requires multiple iterations of elimination, exhibiting a certain degree of randomness in the sparsification process, leading to high computational complexity. Therefore, how to quickly reconstruct the sparse parity-check matrix using received codewords under high bit error rates is crucial. Summary of the Invention

[0004] This invention proposes a method for reconstructing LDPC sparse parity-check matrices under high bit error rates to address the problems in existing technologies, achieving rapid reconstruction of LDPC sparse parity-check matrices under high bit error rates. The technical solution adopted to achieve the above objective is as follows:

[0005] A method for reconstructing a sparse parity-check matrix of LDPC under high bit error rate includes the following steps:

[0006] Step A: Calculate the probability of a single draw containing a verification node, and then determine the number of random draws;

[0007] Step B: Randomly sample the bits of the LDPC codeword to construct a square matrix and perform Gaussian elimination to obtain the suspected check vector;

[0008] Step C: Based on the statistical characteristics of the establishment of suspected check vector relationships and the minimum error decision criterion, determine the suspected check vectors obtained by Gaussian elimination in Step B.

[0009] Step D: Use the check vector retained in step C to make a decision on the overall received codeword matrix, and use the methods of "removing erroneous codewords" or "flipping the least unreliable bit" to increase the proportion of error-free code groups in the received data, depending on the number of received codewords.

[0010] Step E: Repeat steps B through D until the required number of iterations is reached.

[0011] Preferably, in step A, let v' be a sparse parity-check vector in the LDPC sparse parity-check matrix, w' be the number of parity-check nodes, and s be the number of bits randomly extracted from the codeword. Then, for an LDPC with code length n, the number of sample spaces extracted from s positions is: The number of check nodes that exactly contain w' among these s positions is Therefore, the probability that a random draw can include sparse check nodes in v' is:

[0012]

[0013] By using the obtained probability P of randomly selecting a bit containing a sparse check node, the number of random selections is determined, thus reliably ensuring that the selected bits contain a check node. Let the number of random selections be iter. Then, in iter random selections, the number T of times a sparse check node is selected follows a binomial distribution, i.e.

[0014] T~B(iter,P)

[0015] When the number of draws (iter) is large, the De Moivre-Laplace theorem states that...

[0016]

[0017] in This represents a standard normal distribution.

[0018] In mathematical statistics, an event with a probability greater than 0.9975 can be defined as a high-probability event, meaning it occurs at least once during the random sampling of `iter`. Therefore, the range of values ​​for `iter` is [range to be filled in].

[0019]

[0020] Preferably, the LDPC codeword bits are randomly sampled to construct a square matrix, and Gaussian elimination is performed to obtain the suspected verification vector.

[0021] Preferably, step B specifically includes the following steps:

[0022] Step B1: Receive LDPC codeword C N×n A new codeword matrix C' is obtained by randomly selecting bits. N×s ;

[0023] Step B2, for the new codeword matrix C' N×s A square matrix C is obtained by randomly selecting rows. s×s ;

[0024] Step B3, then move on to the square formation C” s×s Gaussian elimination is performed to find the dual space, and then the verification vector is obtained.

[0025] Step B4: Save the solution result from step B3 and send it to step C for judgment.

[0026] Preferably, step B3 specifically includes the following:

[0027]

[0028] Among them I k'×k' Let k'×k' dimensional identity matrix, 0 (s-k')×k' Let C be an (n-k')×k' dimensional matrix containing all zeros. s×s No error code, then D (s-k')×(s-k') If the matrix is ​​all zeros, then the check vector is... All column vectors in C; if C” s×s The presence of errors disrupts some linear relationships, and Gaussian elimination can lead to error propagation. In this case, D... (s-k')×(s-k') Given a sparse matrix, the verification vector is... The middle part of the vector.

[0029] Preferably, step C specifically includes the following:

[0030] Step C1: Calculate the bit error rate p in the channel. e Next, the probability that the verification relationship still holds under the two types of assumptions (H0: h is not a verification vector; H1: h is a verification vector) obtained in step B.

[0031] Step C2: Let Λ be the decision threshold for the two types of hypotheses, and calculate the false alarm probability P for each. f and the probability of missed alarms P a ;

[0032] Step C3: Combining the probabilities of both types of incorrect decisions, solve for the minimum incorrect decision threshold Λ. min ;

[0033] Step C4: Verify the entire received codeword using the h calculated in Step B, and use the difference t between the number of valid and invalid codewords as a statistic. When t ≥ Λ min When this happens, it can be determined as a verification vector.

[0034] Preferably, step C3 specifically includes the following:

[0035]

[0036] In particular, when N is large enough, Λ min It can be approximated as

[0037]

[0038] in

[0039] Preferably, step D specifically includes the following:

[0040] Step D1: Use the check vector obtained in step C to check the entire codeword, and retain the index O of the erroneous codeword. err ;

[0041] Step D2: Determine whether the number of received codewords N is sufficient. Use methods such as "removing erroneous codewords" (N>n) or "flipping the least unreliable bit" (N≤n) to increase the proportion of error-free codewords in the received codewords.

[0042] Preferably, step D2 specifically includes the following:

[0043] Assume the received codeword matrix is ​​C N×n The corresponding bit log-likelihood ratio matrix is ​​Lr N×n The verification vector obtained in step C is h', where N is the number of received codewords and n is the code length.

[0044] When N > n, it is assumed that the number of received codewords is sufficient, and the method of "removing erroneous codewords" is adopted. Based on the erroneous codeword index in step D1, the received codeword matrix C is directly adjusted. N×n By eliminating incorrect codewords and retaining correct codewords, the proportion of error-free codewords in the received codewords is increased.

[0045] When N≤n, it is determined that the number of received codewords is insufficient, and the method of "flipping the least reliable bit" is adopted. The received codeword matrix C is adjusted according to the erroneous codeword index in step D1. N×n Error code C i Detect its corresponding bit reliability | Lr i |, find |Lr in the check bits i The smallest position and the error code C i The bits at the corresponding positions are flipped, i.e.

[0046]

[0047]

[0048] Where J represents the w in the check vector h' in {1,2,…,n}. h' Each unique parity bit position.

[0049] The beneficial effects of this invention are as follows: In the field of channel coding identification, this invention proposes a method for reconstructing an LDPC sparse parity-check matrix under high bit error rate. By randomly sampling codeword bit positions, the dual space dimension is reduced. Furthermore, different processing methods are applied based on the number of received codewords to increase the proportion of error-free code groups in the received data. When the number of received codewords is large (N>n), the parity-check vector obtained by Gaussian elimination is used to back-check the received codewords, eliminating codewords that fail the check (i.e., erroneous codewords), and then iterative identification continues. When the number of received codewords is small (N≤n), for codewords that fail the check, the log-likelihood ratio of the bits is used to find the least reliable bit position and flip its corresponding bit, and then iterative identification continues. Compared with existing algorithms, this algorithm achieves better parity-check matrix reconstruction and recovery performance at the same bit error rate, and has lower algorithm complexity. Attached Figure Description

[0050] Figure 1 This is a flowchart of the present invention;

[0051] Figure 2 This is a diagram showing the comparison of reconstruction rates under different iteration numbers;

[0052] Figure 3 A diagram showing the reconstruction rate comparison under different numbers of codewords;

[0053] Figure 4A diagram showing the reconstruction rate comparison under different numbers of codewords (with insufficient codewords);

[0054] Figure 5 This is a diagram comparing reconstruction rates at different bitrates. Detailed Implementation

[0055] The present invention will now be further described with reference to the accompanying drawings.

[0056] like Figure 1 As shown, a method for reconstructing a sparse parity-check matrix of LDPC under high bit error rate includes the following steps:

[0057] Step A: Calculate the probability of a single sampling containing a validation node, and then determine the number of random samplings. min ;

[0058] Step B: Randomly sample the bits of the LDPC codeword to construct a square matrix and perform Gaussian elimination to obtain the suspected check vector;

[0059] Step C: Based on the statistical characteristics of the establishment of suspected check vector relationships and the minimum error decision criterion, determine the suspected check vectors obtained by Gaussian elimination in Step B.

[0060] Step D: Use the check vector retained in step C to make a decision on the overall received codeword matrix, and use the methods of "removing erroneous codewords" or "flipping the least unreliable bit" to increase the proportion of error-free code groups in the received data, depending on the number of received codewords.

[0061] Step E: Repeat steps B through D until the required number of iterations is reached.

[0062] In step A, the probability of a single sampling containing a check node is calculated, thus determining the number of random samplings. Let v' be a sparse check vector in the LDPC sparse check matrix, corresponding to the number of check nodes w', and s be the number of bits randomly sampled from the codeword. Then, for an LDPC of code length n, the number of sample spaces sampled from s positions is... The number of check nodes that exactly contain w' among these s positions is Therefore, the probability that a random draw can include sparse check nodes in v' is:

[0063]

[0064] By using the obtained probability P1 of randomly selecting bits containing a sparse check node, the number of random selections is determined, thus reliably ensuring that the selected bits contain a check node. Let the number of random selections be iter. Then, in iter random selections, the number T of times a sparse check node can be included follows a binomial distribution, i.e.

[0065] T~B(iter,P1)

[0066] When the number of draws (iter) is large, the De Moivre-Laplace theorem states that...

[0067]

[0068] in This represents a standard normal distribution.

[0069] In mathematical statistics, an event with a probability greater than 0.9975 can be defined as a high-probability event, meaning it occurs at least once during the random sampling of iter. Therefore, obtaining iter must satisfy the following formula.

[0070]

[0071] By consulting the normal distribution table, we can find...

[0072]

[0073] Solving the above equation, we obtain the range of values ​​for iter.

[0074]

[0075] Thus, the minimum number of random samplings required to reliably include sparse check nodes is obtained.

[0076]

[0077] Based on the above, step B specifically includes the following steps:

[0078] Step B1: Receive LDPC codeword C N×n A new codeword matrix C' is obtained by randomly selecting bits. N×s ;

[0079] Step B2, for the new codeword matrix C' N×s A square matrix C is obtained by randomly selecting rows. s×s ;

[0080] Step B3, then move on to the square formation C” s×s Gaussian elimination is performed to find the dual space, and then the verification vector is obtained.

[0081] Step B4: Save the solution result from step B3 and send it to step C for judgment.

[0082] Step B3 specifically includes the following:

[0083]

[0084] Among them I k'×k'Let k'×k' dimensional identity matrix, 0 (s-k')×k' Let C be an (n-k')×k' dimensional matrix containing all zeros. s×s No error code, then D (s-k')×(s-k') If the matrix is ​​all zeros, then the check vector is... All column vectors in C; if C” s×s The presence of errors disrupts some linear relationships, and Gaussian elimination can lead to error propagation. In this case, D... (s-k')×(s-k') Given a sparse matrix, the verification vector is... The middle part of the vector.

[0085] Based on the above, step C specifically includes the following steps:

[0086] Step C1: Calculate the bit error rate p in the channel. e Next, the probability that the verification relationship still holds under the two types of assumptions (H0: h is not a verification vector; H1: h is a verification vector) obtained in step B.

[0087] Step C2: Let Λ be the decision threshold for the two types of hypotheses, and calculate the false alarm probability P for each. f and the probability of missed alarms P a ;

[0088] Step C3: Combining the probabilities of both types of incorrect decisions, solve for the minimum incorrect decision threshold Λ. min ;

[0089] Step C4: Verify the entire received codeword using the h calculated in Step B, and use the difference t between the number of valid and invalid codewords as a statistic. When t ≥ Λ min When this happens, it can be determined as a verification vector.

[0090] Step C3 specifically includes the following:

[0091] First, consider the following two types of assumptions: H0: h is not a check vector; H1: h is a check vector.

[0092] With a channel bit error rate of P e Under the assumption H1, the probability that the verification relationship still holds is:

[0093]

[0094] For the assumption H0, since h is not a check vector, the probability of the codeword check relation being true is random, i.e. The difference t between the number of valid and invalid codewords is used as a statistic. When the number of codewords N is sufficiently large, under assumption H1, t follows a mean of t. variance is The normal distribution, i.e.

[0095]

[0096] Under the assumption H0, t follows a normal distribution with mean 0 and variance N, i.e.

[0097]

[0098] For ease of description, remember Let Λ be the decision threshold for the two types of hypotheses, then the false alarm probability P f for

[0099]

[0100] The probability of missed alarms P a for

[0101]

[0102] Combining the probabilities of two types of incorrect decisions, the average probability of incorrect decision is:

[0103]

[0104] Using P er Differentiating Λ with respect to 0, we get

[0105]

[0106] In particular, when N is large enough, Λ min It can be approximated as

[0107]

[0108] In addition, step D specifically includes the following:

[0109] Step D1: Use the check vector obtained in step C to check the entire codeword, and retain the index O of the erroneous codeword. err ;

[0110] Step D2: Determine whether the number of received codewords N is sufficient. Use methods such as "removing erroneous codewords" (N>n) or "flipping the least unreliable bit" (N≤n) to increase the proportion of error-free codewords in the received codewords.

[0111] Furthermore, in step D2, it is assumed that the received codeword matrix is ​​C. N×n The corresponding bit log-likelihood ratio matrix is ​​Lr N×n The verification vector obtained in step C is h', where N is the number of received codewords and n is the code length.

[0112] When N > n, it is assumed that the number of received codewords is sufficient, and the method of "removing erroneous codewords" is adopted. Based on the erroneous codeword index in step D1, the received codeword matrix C is directly adjusted. N×n By eliminating incorrect codewords and retaining correct codewords, the proportion of error-free codewords in the received codewords is increased.

[0113] When N≤n, it is determined that the number of received codewords is insufficient, and the method of "flipping the least reliable bit" is adopted. The received codeword matrix C is adjusted according to the erroneous codeword index in step D1. N×n Error code C i Detect its corresponding bit reliability | Lr i |, find |Lr in the check bits i The smallest position and the error code C i The bits at the corresponding positions are flipped, i.e.

[0114]

[0115]

[0116] Where J represents the w in the check vector h' in {1,2,…,n}. h' Each unique parity bit position.

[0117] The reconstruction rate of the simulation results is, for example, Figures 2-5 As shown.

[0118] like Figure 2 As shown in the figure, the simulation parameters are set as follows: the experiment uses LDPC codes defined in the IEEE 802.11n protocol. The number of received codewords is set to 3000 (sufficient codeword count), the bit error rate range is 0.0005–0.0055, the number of row iterations is 5, 10, and 20, and the maximum code weight of the parity check vector is set to 10. As can be seen from the figure, the proposed algorithm achieves a higher sparse parity check matrix reconstruction rate than existing algorithms at different iteration counts, and even with a bit error rate of 0.0025, the reconstruction rate of the sparse parity check matrix can reach 100%.

[0119] like Figures 3-4 As shown, the simulation parameters are set as follows: the channel bit error rate is 0.0014, the number of Gaussian elimination iterations is 10, and the number of received LDPC codewords ranges from 498 to 1100 with a value interval of 50. The figure shows that when the number of received codewords is sufficient (N > n), the algorithm in this paper uses "removing erroneous codewords" to improve the reconstruction rate of the sparse parity-check matrix. When the number of received codewords is insufficient (N ≤ n), the algorithm in this paper uses "flipping the least unreliable bit" to mitigate the reduction in reconstruction rate caused by the "removing erroneous codewords" method, and the reconstruction rate is still higher than that of existing algorithms.

[0120] like Figure 5As shown in the figure, the experiment used LDPC codes under the IEEE 802.11n protocol, with a code length of 648, 3000 codewords, 20 row iterations, and code rates of 1 / 2, 2 / 3, and 3 / 4, respectively. The channel bit error rate ranged from 0.0005 to 0.0055. It can be seen from the figure that the proposed algorithm achieves a higher sparse parity-check matrix reconstruction rate than existing algorithms at different code rates.

[0121] For a specific calculation process of the LDPC sparse parity-check matrix reconstruction method under high bit error rate, please refer to the above embodiments. The embodiments of the present invention will not be repeated here.

[0122] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. However, these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for reconstructing a sparse parity-check matrix using LDPC under high bit error rate, characterized in that, Includes the following steps: Step A: Calculate the probability of a single draw containing a verification node, and then determine the number of random draws; Step B: Randomly sample the bits of the LDPC codeword to construct a square matrix and perform Gaussian elimination to obtain the suspected check vector; Step C: Based on the statistical characteristics of the establishment of suspected check vector relationships and the minimum error decision criterion, determine the suspected check vectors obtained by Gaussian elimination in Step B. Step D: The check vector retained in Step C is used to determine the overall received codeword matrix. Depending on the number of received codewords, methods such as "removing erroneous codewords" or "flipping the least reliable bit" are employed to increase the proportion of error-free code groups in the received data. Specifically, this includes the following: Step D1: Use the check vector obtained in step C to check the entire codeword and retain the index of the erroneous codeword. ; Step D2: Determine the number of received codewords. To determine if the code is sufficient, methods such as "removing erroneous codewords" or "flipping the least unreliable bit" can be used to increase the proportion of error-free codewords in the received code group. Specifically: when If the number of received codewords is deemed sufficient, the method of "removing erroneous codewords" is adopted. Based on the erroneous codeword index in step D1, the received codeword matrix is ​​directly processed. By eliminating incorrect codewords and retaining correct codewords, the proportion of error-free codewords in the received codewords is increased. when If the number of received codewords is insufficient, the "flip the least unreliable bit" method is used to adjust the received codeword matrix according to the erroneous codeword index in step D1. Chinese error code Detect its corresponding bit reliability Search in the check bits Minimum position and error codeword The bits at the corresponding positions are flipped; in To receive the number of codewords, For code length; Step E: Repeat steps B through D until the required number of iterations is reached.

2. The method for reconstructing a sparse parity-check matrix under high bit error rate using LDPC according to claim 1, characterized in that, In step A, let a certain sparse check vector in the LDPC sparse check matrix be... The corresponding number of verification nodes is The number of bits randomly selected from the codeword is The code length is LDPC, extraction The number of sample spaces at each location is And this The position contains exactly one The number of verification nodes is Therefore, a single random draw can include The probability of a sparse check node is Using the obtained probability of randomly selecting nodes that include sparse check nodes The number of random samplings is determined to reliably ensure that the sampled bits include a check node; let the number of random samplings be... Then in The number of times a random sample can contain sparse check nodes. It follows a binomial distribution, that is... When the number of draws When the value is large, we can obtain the De Moivre-Laplace theorem. in Represents a standard normal distribution; In mathematical statistics, an event is defined as having a high probability of occurrence when the probability of its occurrence is greater than 0.9975, meaning it is highly probable in random sampling. During the process, it occurred at least once, therefore we get The range of values ​​is 。 3. The method for reconstructing a sparse parity-check matrix under high bit error rate according to claim 2, characterized in that, Randomly sample bits from LDPC codewords to construct a square matrix, then perform Gaussian elimination to obtain a potential check vector.

4. The LDPC sparse parity-check matrix reconstruction method under high bit error rate according to claim 3, characterized in that, Step B specifically includes the following steps: Step B1: Receive LDPC codewords New codeword matrix obtained by randomly selecting bits ; Step B2: Apply the new codeword matrix Obtain a square matrix by randomly selecting rows. ; Step B3, then proceed to the opponent's formation Gaussian elimination is performed to find the dual space, thereby obtaining the suspected verification vector; Step B4: Save the solution result from step B3 and send it to step C for judgment.

5. The LDPC sparse parity-check matrix reconstruction method under high bit error rate according to claim 4, characterized in that, Using Gaussian elimination over a binary field Elementary transformations are performed to convert it into an upper triangular matrix. Step B3 specifically includes the following: in, for An identity matrix of dimensions for A zero matrix of dimension; if No errors, then If the matrix is ​​all zeros, then the check vector is... All column vectors in; if The presence of bit errors disrupts some linear relationships, leading to bit error propagation during Gaussian elimination. Given a sparse matrix, the verification vector is... The middle part of the vector.

6. A method for reconstructing a sparse parity-check matrix under high bit error rate according to any one of claims 1 to 5, characterized in that, Step C specifically includes the following steps: Step C1, set For the suspected check vector obtained in step B, calculate the bit error rate at the channel bit error rate. Under the following assumptions for Not a suspected verification vector, or an assumption. for When the vector is a suspected check vector, the probability that the check relationship is true. ; Step C2: Let the decision threshold for the two types of hypotheses be... Calculate the false alarm probability respectively and the probability of missed alarms ; Step C3: Combining the probabilities of both types of incorrect decisions, solve for the minimum incorrect decision threshold. ; Step C4, Solved using the solution from step B The entire received codeword is verified, and the difference between the number of valid and invalid codewords is calculated. As a statistic; when When this happens, it can be determined as a verification vector.

7. The method for reconstructing a sparse parity-check matrix under high bit error rate using LDPC according to claim 6, characterized in that, In step C3, the minimum error decision threshold is satisfied as follows: when When the large sample condition is met, Approximately: 。 8. The method for reconstructing a sparse parity-check matrix under high bit error rate according to claim 1, characterized in that, In step D2, let the bit log-likelihood ratio matrix be... The verification vector obtained in step C is ,in To receive the number of codewords, The code length; when flipping the least unreliable bit, search in the parity bit. Minimum position and error code The bits at the corresponding positions are flipped as follows: ； in express Middle verification vector middle Each unique parity bit position.

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