Complexity reduction method based on RM code recursive projection aggregation decoding
By selectively retaining specific projection subspace in recursive projection aggregation decoding and using verification information to judge the iteration to terminate, combined with the attenuation factor, the complexity of recursive projection aggregation decoding of RM code is reduced, the problem of high complexity is solved, and an efficient decoding process is achieved.
Patent Information
- Application Number
- CN202510453756.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-04-11
- Publication Date
- 2025-07-25
AI Technical Summary
The complexity of recursive projection aggregation decoding is too high, which limits its practical application in communication systems.
In the recursive projection process, the specific projection subspace is selectively retained, the likelihood ratio information is updated through iterative decoding, and the decoding termination is judged by using the verification information, and the number of times of fast Hadamar transformation is reduced with the attenuation factor.
On the premise of retaining error correction capabilities, the complexity of recursive projection aggregation decoding is significantly reduced, the operating cost is reduced, and the decoding efficiency is improved.
Smart Images
Figure CN120377937A_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the field of information and communication technologies, and particularly relates to a method for reducing complexity based on recursive projection aggregation decoding of RM codes. Background Art
[0002] The development of wireless communication has led to an increasing demand for the reliability and efficiency of data transmission, and channel coding technology has also become an essential part of the data transmission process. As one of the oldest codes, RM codes have a simple construction structure, and the special structure of its first-order RM codes can achieve fast maximum likelihood decoding. For a long time, the academic community has speculated that RM codes can achieve the maximum channel capacity. This conjecture was proven to be true in 2017. The breakthrough achieved by recursive projection aggregation decoding is expected to trigger a new round of research on RM codes by domestic and foreign scholars, bringing new opportunities to the currently booming theory and practice of coding and decoding. To make up for the extremely high complexity of recursive projection aggregation decoding caused by the complex projection process, the following two strategies can be adopted. One strategy is recursive penetration. Another strategy is to find a suitable method to reduce the large number of projection subspaces generated during the decoding process.
[0003] Whether adopting penetration or reducing the projection space at each level, it is necessary to comprehensively consider information such as code length, order, and number of projections as a prerequisite. This leads to the need to find an appropriate method to reduce projections. By using a reasonable method to reduce the projection subspace, the problems caused by the excessively high complexity of recursive projection aggregation decoding can be compensated to a certain extent, which is beneficial to the successful practical application of recursive projection aggregation decoding in communication systems in the future and significantly reduces the high operating cost brought by using recursive projection aggregation decoding. Therefore, reducing complexity is of great significance for the successful application of recursive projection aggregation decoding in the operation of communication systems. Summary of the Invention
[0004] To achieve the above object, the present invention provides a method for reducing complexity based on recursive projection aggregation decoding of RM codes, including:
[0005] Selectively retaining specific projection subspaces during the recursive projection process, and updating the likelihood ratio information based on iterative decoding according to the specific projection subspaces;
[0006] Judging the termination of iteration of decoding according to the parity check information and outputting the decoding result.
[0007] Preferably, the process of selectively retaining specific projection subspaces during the recursive projection process includes:
[0008] Determining the specific projection subspaces to be retained based on the uniqueness of the projection subspaces during the recursive projection aggregation decoding process of RM codes to obtain non-repeated projection subspaces.
[0009] Preferably, after the specific projection subspace, the process of updating the likelihood ratio information based on iterative decoding includes:
[0010] Calculate the number of unique subspaces according to the formula, and according to the law of repetition phenomenon, retain the specific projection subspace and perform the first decoding of the decoding iteration.
[0011] Then, after retaining the specific projection subspace, perform subsequent decoding iterations, and only retain part of the projection subspaces in each subsequent iteration.
[0012] Preferably, the formula expression for calculating the number of unique subspaces according to the formula is:
[0013]
[0014] Where, for RM(m,r), r>2, N = 2 r-1 Is the number of projection subspaces in the recursive projection aggregation decoding process, N D Represents the number of repeated projection subspaces, and m represents the number of variables of the generating polynomial involved in encoding when constructing the RM code.
[0015] Preferably, the process of retaining the specific projection subspace and performing the first decoding of the decoding iteration according to the law of repetition phenomenon includes:
[0016] In the recursive projection aggregation decoding process, after two recursive projections, a characteristic repetition phenomenon occurs when the recursive level f = r - 2. Define a projection subspace Corresponding to a repeated structure that will be generated after two recursive projections. For the non-repeated part, it is as follows:
[0017]
[0018] Where, b represents the subspace position order generated in the first-level recursive projection process.
[0019] Preferably, the process of performing subsequent decoding iterations after retaining the specific projection subspace and only retaining part of the projection subspaces in each subsequent iteration includes:
[0020] Define the attenuation factor d >= 1, use the attenuation factor to retain 1 / d of the projection subspaces in the iteration, and further apply it to each level of recursion in the algorithm. In each iteration of each level of recursion, only retain 1 / d of the number of projection subspaces retained in the previous iteration.
[0021] Preferably, the process of judging the termination iteration of decoding according to the parity check information and outputting the decoding result includes:
[0022] The likelihood ratio information obtained by iterative decoding should be judged and then multiplied by the parity-check matrix. If the result is zero, it is determined that the legitimate codeword stops iterating;
[0023] The likelihood ratio information obtained after iteration is processed and continuously aggregated and updated to finally output the decoding information.
[0024] Preferably, the process that the likelihood ratio information obtained by iterative decoding should be judged and then multiplied by the parity-check matrix, and if the result is zero, it is determined that the legitimate codeword stops iterating includes:
[0025] Define the codeword received through channel transmission as x. Assume that the legitimate codeword input to the channel is c. Multiply c by the parity-check matrix H and x. If the result is equal to 0, then x is called a legitimate codeword;
[0026] For RM(m,r), the parity-check matrix is H = G(m,m - r - 1), where G(m,m - r - 1) is the generating matrix of RM(m,m - r - 1);
[0027] Use the multiplication result of the parity-check matrix of RM(m,r) code as the syndrome check index to determine whether the iteration terminates;
[0028] The condition for determining whether the iteration terminates is:
[0029] Given the received log-likelihood ratio vector L, perform syndrome checking to check whether where represents the iterative likelihood ratio information obtained when the iteration is completed.
[0030] Preferably, the process that the likelihood ratio information obtained after iteration is processed and continuously aggregated and updated to finally output the decoding information includes:
[0031] Recursive projection aggregation decoding of RM code. Take out an s-dimensional subspace of the codewords generated by RM(m,r) in the m-dimensional binary vector space for projection;
[0032] The projection of RM(m,r) on the subspace is the subcode RM(m - s,r - s). For the first projection, s = 1. Excluding the all-0 one-dimensional subspace, there are 2 m - 1 one-dimensional subspaces, and 2 m - 1 projection results are obtained;
[0033] Continue to project RM(m - s,r - s) until the first-order RM(m - r + 1,1) code. The first-order RM code uses fast Hadamard transform for decoding;
[0034] After obtaining the iterative likelihood ratio information, the likelihood ratio is aggregated and updated upward. The input likelihood ratio of the previous level is corrected step by step according to the likelihood ratio of the recursive codewords. After the last-level aggregation, a new round of recursive projection aggregation decoding is started. Among them, r represents the order of the RM code, and m represents the number of variables of the generating polynomial involved in the encoding when constructing the RM code.
[0035] Preferably, the formula expression for aggregating and updating the likelihood ratio upward is:
[0036]
[0037] Among them, the decoded codeword i represents the i-th subset of the projection, and z i represents the coset corresponding to the i-th subset B i generated by the projection, and L cumu refers to the total likelihood ratio information after aggregating all subsets, refers to the likelihood ratio information after iterative update.
[0038] Compared with the prior art, the present invention has the following advantages and technical effects:
[0039] The present invention realizes the reduction of the complexity of recursive projection aggregation decoding. In each level of iteration process, a unique projection subspace and an attenuation factor are adopted to reduce the number of fast Hadamard transforms. Since the recursive projection aggregation decoding corrects the error information generated during the transmission process, which mostly occurs in the first iteration decoding process, and by using a special unique projection subspace, a new idea is provided to minimize the number of times of using the fast Hadamard transform on the premise of completely retaining the error correction ability of the first iteration. And by combining the functions of adding the parity-check matrix and the attenuation factor, the purpose of further reducing the complexity of recursive projection aggregation decoding is achieved. BRIEF DESCRIPTION OF THE DRAWINGS
[0040] The drawings constituting a part of this application are used to provide a further understanding of this application. The schematic embodiments of this application and their descriptions are used to explain this application and do not constitute an improper limitation to this application. In the drawings:
[0041] Figure 1 is a schematic flowchart of the method according to an embodiment of the present invention;
[0042] Figure 2 is a schematic diagram of the decoding structure according to an embodiment of the present invention;
[0043] Figure 3 is a schematic diagram of the simulation result according to an embodiment of the present invention;
[0044] Figure 4 is a schematic diagram of the theoretical calculation according to an embodiment of the present invention;
[0045] Figure 5 Schematic diagram of the system structure according to an embodiment of the present invention. Detailed implementation manners
[0046] It should be noted that, without conflict, the embodiments in the present application and the features in the embodiments may be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in conjunction with the embodiments.
[0047] It should be noted that the steps shown in the flowchart of the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although the logical order is shown in the flowchart, in some cases, the steps shown or described can be executed in a different order than here.
[0048] Embodiment 1
[0049] As Figure 1-2 shown, a method for reducing complexity based on recursive projection aggregation decoding of RM codes is provided in this embodiment, including the following steps:
[0050] S1. Selectively retain specific projection subspaces during the recursive projection process;
[0051] Specifically, based on the uniqueness of the projection subspaces during the recursive projection aggregation decoding process of RM codes, the specific projection subspaces to be retained are determined to obtain non-repeated projection subspaces.
[0052] Furthermore, only the non-repeated projection subspaces in the recursive projection aggregation decoding are used during the iteration process. For RM(m,r), in order to achieve the purpose of selecting non-repeated projections during the recursive projection aggregation decoding process when r>2, it is necessary to first determine the number of repeated projections when r>2, N = 2 r-1 is the number of projection subspaces during the recursive projection aggregation decoding process, N D represents the number of repeated projection subspaces, and m represents the number of variables of the generating polynomial involved in the encoding when constructing RM codes;
[0053]
[0054] During the recursive projection aggregation decoding process, after two recursive projections, a characteristic repetition phenomenon will occur when the recursive level f = r - 2. Define a projection subspace corresponding to a repeated structure generated after two recursive projections. For the non-repeated part, it is as follows:
[0055]
[0056] Among them, b represents the order of the subspace positions generated during the first-level recursive projection process.
[0057] S2. After retaining the specific projection subspace, update the likelihood ratio information based on iterative decoding;
[0058] Specifically, calculate the number of unique subspaces according to the formula, and retain the specific projection subspace according to the law of repetition phenomena for the first decoding of the decoding iteration;
[0059] Then, after retaining the specific projection subspace, perform subsequent decoding iterations, and only retain part of the projection subspaces in each subsequent iteration.
[0060] Furthermore, recursive projection aggregation decoding uses this iterative method in the process of each-level recursive projection. Define the attenuation factor d >= 1 and use it to retain 1 / d of the projection subspaces in the iteration. This reduction method is applied to each level of recursion in the algorithm, and in each iteration of each level of recursion, only 1 / d of the number of projection subspaces retained in the previous iteration will be retained.
[0061] S3. Judge the termination of the decoding iteration according to the parity-check information and output the decoding result;
[0062] Specifically, multiply the likelihood ratio information obtained by iterative decoding by the parity-check matrix for judgment. If the result is zero, it is judged as a legitimate codeword and the iteration stops;
[0063] Process the likelihood ratio information obtained after iteration and continuously aggregate and update to finally output the decoding information.
[0064] Furthermore, define the codeword received through the channel as x, assume the legitimate codeword input to the channel is c, and multiply it by the parity-check matrix H and x. If the result is equal to 0, then x is called a legitimate codeword. For RM(m,r), the parity-check matrix is H = G(m,m - r - 1), where G(m,m - r - 1) is the generator matrix of RM(m,m - r - 1). In this work, the result of multiplying by the parity-check matrix of the RM(m,r) code is used as the syndrome check index to judge when to terminate the iteration. Specifically, given the received log-likelihood ratio vector L, perform a syndrome check to check whether where represents the iterative likelihood ratio information obtained when the iteration is completed.
[0065] For the recursive projection aggregation decoding of the RM code, take out an s-dimensional subspace (usually take s = 1) of the codeword generated by RM(m,r) in the m-dimensional binary vector space for projection. The projection of RM(m,r) on the subspace is its subcode RM(m - s,r - s). For the first projection (s = 1), there are 2 m - 1 one-dimensional subspaces (excluding the all-zero one-dimensional subspace), which means there are 2 m-1 projection result. Project RM(m-s, r-s) continuously until the first-order RM(m-r+1, 1) code, and the first-order RM code is decoded using the fast Hadamard transform.
[0066] After obtaining the iterative likelihood ratio information, perform upward aggregation update (opposite to the recursive projection aggregation decoding that projects from high dimension to low dimension, update the corresponding high-dimensional projection information from several low-dimensional projections), and gradually correct the input likelihood ratio of the previous level according to the likelihood ratio of the recursive codeword (aggregation operation). After the last-level aggregation, start a new round of recursive projection aggregation decoding, where r represents the order of the RM code, and m represents the number of variables of the generating polynomial involved in the encoding when constructing the RM code.
[0067] The calculation method for aggregating and updating the likelihood ratio includes:
[0068]
[0069] Among them, the decoded codeword i represents the i-th subset of the projection, and z i represents the coset corresponding to the i-th subset B i generated by the projection, and L cumu refers to the total likelihood ratio information after aggregating all subsets, refers to the likelihood ratio information after iterative update.
[0070] Such as Figure 3 and Figure 4 shown, which are the simulation results of this embodiment, comparing the method of reducing complexity of recursive projection aggregation decoding and the way of reducing complexity proposed by the present invention. As can be seen from the figure, the bit error rate of the recursive projection aggregation that retains all projections has no significant difference from the performance after reducing complexity.
[0071] Embodiment 2
[0072] Such as Figure 5 shown, based on the same inventive concept, this embodiment also provides a complexity-reduced decoding system based on recursive projection aggregation decoding of RM codes, including: a unique projection subspace determination module, an attenuation module, a check module, and an update module;
[0073] Among them, the unique projection subspace determination module is used to selectively retain specific projection subspaces during the recursive projection process;
[0074] Specifically, based on the uniqueness of the projection subspaces during the recursive projection aggregation decoding process of RM codes, determine the specific projection subspaces to be retained to obtain non-repeated projection subspaces.
[0075] The unique projections that occur during the projection process of recursive projection aggregation decoding include:
[0076] Only non-repeated projection subspaces in recursive projection aggregation decoding are used during the iteration process. For RM(m, r), to select non-repeated projections during the recursive projection aggregation decoding process when r > 2, it is necessary to first determine the number of repeated projections when r > 2, N = 2 r-1 is the number of projection subspaces in the recursive projection aggregation decoding process, N D represents the number of repeated projection subspaces, and m represents the number of variables of the generating polynomial involved in encoding when constructing the RM code;
[0077]
[0078] During the recursive projection aggregation decoding process, after two recursive projections, that is, when f = r - 2, a characteristic repetition phenomenon will occur. Define a projection subspace corresponding to a repeated structure generated after two recursive projections. For the non-repeated part, it is as follows:
[0079]
[0080] where b represents the order of the subspace positions generated during the first-level recursive projection process.
[0081] The attenuation module is used to retain specific projection subspaces and then update the likelihood ratio information based on iterative decoding. Specifically, calculate the number of unique subspaces according to the formula, and retain specific projection subspaces according to the repetition phenomenon rule for the first decoding of the decoding iteration;
[0082] Then, after retaining specific projection subspaces, perform subsequent decoding iterations, and only retain part of the projection subspaces in each subsequent iteration.
[0083] Furthermore, it includes: Recursive projection aggregation decoding uses the iterative method in the process of each-level recursive projection. Define the attenuation factor d >= 1 and use it to retain 1 / d of the projection subspaces during the iteration. This reduction method is applied to each level of recursion in the algorithm, and in each iteration of each level of recursion, only 1 / d of the number of projection subspaces retained in the previous iteration is retained.
[0084] The parity-check module and the update module are used to judge the termination of the decoding iteration according to the parity-check information and output the decoding result;
[0085] Specifically, multiply the likelihood ratio information obtained by iterative decoding by the parity-check matrix for judgment. If the result is zero, judge the legal codeword and stop the iteration;
[0086] The likelihood ratio information obtained after iteration is processed and continuously aggregated and updated to finally output the decoding information.
[0087] The working process of the parity-check module is as follows:
[0088] Define the received codeword transmitted through the channel as \(x\). Assume that the legitimate codeword input to the channel is \(c\). Multiply \(x\) by the parity-check matrix \(H\). If the result is equal to \(0\), then \(x\) is called a legitimate codeword. For \(RM(m,r)\), the parity-check matrix is \(H = G(m,m - r - 1)\), where \(G(m,m - r - 1)\) is the generator matrix of \(RM(m,m - r - 1)\). In this work, the result of multiplying by the parity-check matrix of the \(RM(m,r)\) code is used as a syndrome test metric to determine when to terminate the iteration. Specifically, given the received log-likelihood ratio vector \(L\), perform a syndrome check to check whether where represents the iterative likelihood ratio information obtained when the iteration is completed.
[0089] The update module includes:
[0090] Recursive projection aggregation decoding of the RM code. Take a \(s\)-dimensional subspace (usually \(s = 1\)) of the codewords generated by \(RM(m,r)\) in the \(m\)-dimensional binary vector space for projection. The projection of \(RM(m,r)\) on the subspace is its subcode \(RM(m - s,r - s)\). For the first projection (\(s = 1\)), there are \(2 m - 1 one-dimensional subspaces (excluding the all-zero one-dimensional subspace), which means there are \(2 m - 1 projection results. Project \(RM(m - s,r - s)\) continuously until the first-order \(RM(m - r + 1,1)\) code, and the first-order RM code is decoded using the fast Hadamard transform.
[0091] After obtaining the iterative likelihood ratio information, perform upward aggregation update (opposite to the recursive projection aggregation decoding from high-dimensional to low-dimensional projection, update the corresponding high-dimensional projection information from several low-dimensional projections), and gradually correct the input likelihood ratio of the previous level according to the likelihood ratio of the recursive codewords (aggregation operation). After the last-level aggregation, start a new round of recursive projection aggregation decoding, where \(r\) represents the order of the RM code, and \(m\) represents the number of variables of the generating polynomial involved in encoding when constructing the RM code.
[0092] The calculation method for aggregating and updating the likelihood ratio includes:
[0093]
[0094] where the decoded codeword \(i\) represents the \(i\)-th subset of the projection, \(z i represents the coset corresponding to the \(i\)-th subset \(B i generated by the projection, \(L cumu refers to the total likelihood ratio information after aggregating all subsets, refers to the likelihood ratio information after iterative update.
[0095] The complexity reduction system based on recursive projection aggregation decoding of RM codes provided in this embodiment has all the advantages of the complexity reduction method based on recursive projection aggregation decoding of RM codes provided in Embodiment 1.
[0096] Embodiment 3
[0097] This embodiment also discloses a computer device, including a memory, a processor, and a computer program stored on the memory. The processor executes the computer program to implement the steps of the method described in Embodiment 1.
[0098] Embodiment 4
[0099] This embodiment also discloses a computer-readable storage medium, on which a computer program is stored. When the computer program is executed by a processor, it implements the steps of the method described in Embodiment 1.
[0100] Embodiment 5
[0101] This embodiment also discloses a computer program product, including a computer program. When the computer program is executed by a processor, it implements the steps of the method described in Embodiment 1.
[0102] The above is only a preferred specific embodiment of the present application, but the protection scope of the present application is not limited thereto. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed in the present application should be covered by the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.
Claims
1. A method for reducing complexity based on recursive projection aggregation decoding of RM codes, characterized in that Including: During the recursive projection process, selectively retain a specific projection subspace, and based on the specific projection subspace, update the likelihood ratio information based on iterative decoding. Judge the termination iteration of decoding according to the parity check information and output the decoding result.
2. The method according to claim 1, characterized in that The process of selectively retaining a specific projection subspace during the recursive projection process includes: Based on the uniqueness of the projection subspace in the recursive projection aggregation decoding process of the RM code, determine the retained specific projection subspace to obtain a non-repeated projection subspace.
3. The method according to claim 1, characterized in that After the specific projection subspace, the process of updating the likelihood ratio information based on iterative decoding includes: Calculate the number of unique subspaces according to the formula, and according to the law of repetition phenomenon, retain the specific projection subspace and perform the first decoding of the decoding iteration. Then, after retaining the specific projection subspace, perform subsequent decoding iterations, and only retain a part of the projection subspaces in each subsequent iteration.
4. The method according to claim 3, characterized in that The formula expression for calculating the number of unique subspaces according to the formula is: Among them, for RM(m, r), r > 2, N = 2 r-1 is the number of projection subspaces in the recursive projection aggregation decoding process, N D represents the number of repeated projection subspaces, and m represents the number of variables of the generating polynomial involved in encoding when constructing the RM code.
5. The method according to claim 3, characterized in that The process of retaining the specific projection subspace according to the law of repetition phenomenon and performing the first decoding of the decoding iteration includes: In the process of recursive projection aggregation decoding, after two recursive projections, a characteristic repetition phenomenon occurs when the recursive level f = r - 2, and a projection subspace is defined Corresponding to a repeated structure generated by two recursive projections, the non-repeated part is as follows: where b represents the subspace position order generated in the first-level recursive projection process.
6. The method according to claim 3, characterized in that After retaining the specific projection subspace, the process of performing subsequent decoding iterations and only retaining a part of the projection subspaces in each subsequent iteration includes: Define the attenuation factor d>=1, use the attenuation factor to retain 1 / d of the projection subspaces in the iteration, and further apply it to each level of recursion in the algorithm. In each iteration of each level of recursion, only retain 1 / d of the number of projection subspaces retained in the previous iteration.
7. The method according to claim 1, characterized in that The process of judging the termination iteration of decoding according to the parity check information and outputting the decoding result includes: Multiply the likelihood ratio information obtained by iterative decoding by the parity check matrix after judgment. If the result is zero, judge the legal codeword and stop the iteration; Process the likelihood ratio information obtained after iteration and continuously aggregate and update to finally output the decoding information.
8. The method according to claim 7, characterized in that The process of multiplying the likelihood ratio information obtained by iterative decoding by the parity check matrix after judgment and stopping the iteration if the result is zero includes: Define the codeword received through the channel as x, assume the channel input legal codeword is c, multiply it by the parity check matrix H and x. If the result is equal to 0, then x is called a legal codeword; For RM(m,r), the parity check matrix is H = G(m,m-r-1), where G(m,m-r-1) is the generating matrix of RM(m,m-r-1); Use the result of multiplying by the parity check matrix of RM(m,r) code as the syndrome test index to judge whether the iteration terminates; The condition for judging whether the iteration terminates is: Given the received log-likelihood ratio vector L, syndrome checking is performed to check whether where represents the iterative likelihood ratio information obtained when the iteration is completed.
9. The method according to claim 7, characterized in that The process of processing the likelihood ratio information obtained after iteration and continuously aggregating and updating the final output decoding information includes: RM code recursive projection aggregation decoding, taking the codeword generated by RM(m,r) in the m-dimensional binary vector space and projecting it into an s-dimensional subspace; The projection of RM(m,r) on the subspace is the subcode RM(m - s,r - s). For the first projection, s = 1. Excluding the all-zero one-dimensional subspace, there are 2 m - 1 one-dimensional subspaces, obtaining 2 m - 1 projection results; Continue to project RM(ms,rs) until the first-order RM(m-r+1,1) code is obtained. The first-order RM code is decoded using the fast Hadamard transform. After obtaining the iterative likelihood ratio information, the likelihood ratio is updated by upward aggregation, and the input likelihood ratio of the previous level is corrected step by step according to the likelihood ratio of the recursive codeword. After the last level of aggregation, a new round of recursive projection aggregation decoding begins; where r represents the order of the RM code, and m represents the number of variables of the generating polynomial involved in the encoding when constructing the RM code.
10. The method according to claim 9, characterized in that The formula expression of the upward aggregation updating likelihood ratio is: Among them, the decoded codeword i represents the i-th subset of the projection, z i represents the co-set corresponding to the i-th subset B i produced by the projection, and L cumu refers to the total likelihood ratio information after aggregating all subsets, and refers to the likelihood ratio information after iterative update.