Binary linear code generation matrix reduction method and device based on lattice basis reduction algorithm
By proposing a binary linear code generator matrix reduction method based on the DeepLLL lattice reduction algorithm, the problem of low efficiency in linear code generator matrix reduction algorithms is solved, and efficient generator matrix reduction and balanced Hamming weight distribution are achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- NO 30 INST OF CHINA ELECTRONIC TECH GRP CORP
- Filing Date
- 2024-11-26
- Publication Date
- 2026-06-16
AI Technical Summary
Existing algorithms for reducing linear code generation matrices are inefficient, leading to increased computational and storage costs.
The DeepLLL reduction algorithm based on lattice basis is adopted. By calculating the Epipodal vector in the Epipodal matrix, Lovasz condition judgment and XOR operation of basis vectors, insertion and reduction operations are performed to optimize the reduction process of the generated matrix.
It improves the reduction efficiency of the generated matrix, balances the Hamming weight distribution of the Epipodal vector, and reduces computational and storage costs.
Smart Images
Figure CN119906441B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of data processing technology, and in particular to a method and apparatus for reducing a binary linear code generation matrix based on the DeepLLL lattice reduction algorithm. Background Technology
[0002] In recent years, with the continuous development of communication technology, the reduction algorithm of the generator matrix of linear codes has become an important component in the fields of information transmission and data storage. This is mainly because the reduction algorithm of the generator matrix can greatly optimize the efficiency of the encoding and decoding process, and also helps to design more efficient and resource-utilizing communication systems. The generator matrix of a linear code is a common form used to describe the characteristics of the linear code itself, and can be constructed by adding the required row or column vectors to an identity matrix. However, the generator matrix itself becomes more complex as the amount of data in the communication system increases, leading to a sharp increase in computational and storage costs. Therefore, how to optimize the reduction algorithm of the generator matrix of linear codes has become an urgent problem to be solved in the field of information transmission. Summary of the Invention
[0003] This invention provides a method and apparatus for reducing binary linear code generation matrices based on the DeepLLL reduction algorithm, in order to solve the problem of low processing efficiency of existing linear code generation matrix reduction algorithms.
[0004] In a first aspect, the present invention provides a method for reducing a binary linear code generation matrix based on the lattice-based DeepLLL reduction algorithm, the method comprising:
[0005] Given an arbitrary generator matrix of a binary linear code, calculate the corresponding Epipodal matrix and each Epipodal vector in the Epipodal matrix.
[0006] Determine any Epipodal vector in the Epipodal matrix Is it the shortest non-zero Epipodal vector under the i-th orthogonal projection? If not, it means that there exists a basis vector. Its length under the i-th orthogonal projection is shorter. Perform Lovasz conditional judgment on the two basis vectors. If it is true, output the generator matrix and its corresponding Epipodal matrix.
[0007] Perform a Lovasz condition check on two basis vectors; if satisfied, then apply the Lovasz condition to the basis vectors. Perform an XOR operation, then proceed to the basis vectors that satisfy the reduction condition. Perform a basis reduction operation; otherwise, directly perform a basis reduction operation on the basis vectors that satisfy the reduction condition.
[0008] Perform basis reduction operations on the basis vectors that satisfy the reduction conditions, reducing them to the basis vectors formed by... An element in the generated basic field;
[0009] For basis vectors The insertion operation is performed, and the generated matrix and its corresponding Epipodal matrix are updated in real time. Then, it is determined whether any Epipodal vector in the Epipodal matrix is the shortest non-zero Epipodal vector under the i-th orthogonal projection.
[0010] Optionally, determine any Epipodal vector in the Epipodal matrix. Is it the shortest non-zero Epipodal vector corresponding to the i-th orthogonal projection, including:
[0011] The original basis vectors will be ranked After all the basis vectors are subjected to the same orthogonal projection in sequence, the original basis vectors are compared with their subsequent orthogonal projections in pairs to find the shortest non-zero Epipodal vector under each orthogonal projection, so as to make the Hamming weight distribution of each Epipodal vector more balanced.
[0012] Optionally, the step of inputting an arbitrary generator matrix of a binary linear code, calculating the corresponding Epipodal matrix and each Epipodal vector in the Epipodal matrix, includes:
[0013] Enter one Binary linear code Any "correct" generating matrix The generator matrix is generated by linearly independent Composed of dimensional vectors, denoted in sequence as follows: ,in , These are the relevant initial parameters of the linear code; "correct" means that... The Epipodal vectors corresponding to the basis vectors are all non-zero vectors;
[0014] Calculate the basis vectors sequentially The corresponding Epipodal vector , where basis vectors The corresponding Epipodal vector is itself, that is... The remaining Epipodal vectors can be calculated.
[0015] The calculated Epipodal vectors are used as row vectors to form a generator matrix. The corresponding Epipodal matrix .
[0016] Optionally, determining whether any Epipodal vector in the Epipodal matrix is the shortest non-zero Epipodal vector under the corresponding i-th orthogonal projection further includes:
[0017] Initialization parameters ,calculate ;
[0018] Determine the first inequality Whether it is valid, among which It refers to the first The effect under orthographic projection This refers to the bitwise XOR operation between vectors. If the first inequality shown is true, then the Lovasz condition is checked on the two basis vectors; otherwise, the calculation is performed. And further judgment If k+1 holds true, then further determine whether the first inequality holds true. Is it true? Otherwise, calculate. ,judge If k holds true, then calculate... Conversely, output the real-time generator matrix in the algorithm at this point. and its corresponding Epipodal matrix ;
[0019] in, .
[0020] Optionally, the Lovasz conditional judgment on the two basis vectors includes:
[0021] For those that make the first equation not true Two basis vectors that hold true , Perform Lovasz condition checks;
[0022] Determine the second inequality Whether it is valid, among which It is a piecewise function, for ,like If it is an odd number, then ;like ,but ,in that is, vector The index of the first non-zero element; otherwise, If the second inequality holds, then calculate .
[0023] Optionally, the basis vectors satisfying the reduction condition are reduced to basis reduction operations, reducing them to the form given by... An element in the generated basic field includes:
[0024] Initialization parameters ;
[0025] Determine the second inequality Does it hold true? If the second inequality holds true, then calculate... Conversely, calculate and in Then, continue to determine whether the second inequality holds true.
[0026] Optionally, the insertion operation on the basis vector includes:
[0027] basis vectors Insert into basis vectors and Between, from the basis vectors to basis vectors All basis vectors are shifted one position to the right in sequence, i.e. ;
[0028] In a second aspect, the present invention provides a binary linear code generation matrix reduction device based on the lattice-based DeepLLL reduction algorithm, the device comprising:
[0029] The first processing module is used to take an arbitrary generator matrix of a binary linear code as input, and calculate the Epipodal matrix corresponding to the generator matrix and each Epipodal vector in the Epipodal matrix.
[0030] The judgment module is used to determine whether any Epipodal vector in the Epipodal matrix is the shortest non-zero Epipodal vector under the corresponding i-th orthogonal projection. If not, the Lovasz condition is checked on the two basis vectors. If yes, the generated matrix and its corresponding Epipodal matrix are output. The Lovasz condition is checked on the two basis vectors. If the condition is satisfied, the basis vectors are XORed, and then the basis reduction operation is performed on the basis vectors that satisfy the reduction condition. Otherwise, the basis reduction operation is directly performed on the basis vectors that satisfy the reduction condition. The basis reduction operation is then performed on the basis vectors that satisfy the reduction condition, reducing them to an element in the generated basic field.
[0031] The second processing module is used to perform insertion operations on the basis vectors, update the generated matrix and its corresponding Epipodal matrix in real time, and then determine whether any Epipodal vector in the Epipodal matrix is the shortest non-zero Epipodal vector under the corresponding i-th orthogonal projection.
[0032] Optionally, the judgment module is further configured to perform the same orthogonal projection on all basis vectors following the original basis vector in sequence, and then compare the original basis vector with the subsequent orthogonal projections in pairs to find the shortest non-zero Epipodal vector under each orthogonal projection, so as to make the Hamming weight distribution of each Epipodal vector more balanced.
[0033] Thirdly, the present invention provides a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the binary linear code generation matrix reduction method based on the lattice-based DeepLLL reduction algorithm described above.
[0034] The beneficial effects of this invention are as follows:
[0035] In the selection operation of whether the Epipodal vector corresponding to any basis vector in a given generator matrix is the shortest non-zero vector under its corresponding orthogonal projection, this invention performs the same orthogonal projection on all basis vectors after the original basis vector and then compares them in pairs, thereby finding the shortest non-zero Epipodal vector under each orthogonal projection, making the Hamming weight distribution of each Epipodal vector more balanced. At the same time, performing a reduction operation on the basis vector that meets the relevant conditions can transform the basis vector into an element in the fundamental domain generated by a submatrix of the generator matrix, thereby further optimizing the Hamming weight of the basis vector. In the basis vector insertion operation, for the basis vector that has met the reduction conditions, it only needs to be directly inserted into the corresponding order position, instead of exchanging it with the basis vectors before it, thereby improving the efficiency of the calculation in the update matrix stage.
[0036] The above description is merely an overview of the technical solution of the present invention. In order to better understand the technical means of the present invention and to implement it in accordance with the contents of the specification, and in order to make the above and other objects, features and advantages of the present invention more apparent and understandable, specific embodiments of the present invention are described below. Attached Figure Description
[0037] Various other advantages and benefits will become apparent to those skilled in the art upon reading the following detailed description of preferred embodiments. The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. Furthermore, the same reference numerals denote the same parts throughout the drawings. In the drawings:
[0038] Figure 1 This is a flowchart illustrating a binary linear code generation matrix reduction method based on the DeepLLL reduction algorithm provided in an embodiment of the present invention.
[0039] Figure 2 This is a schematic diagram of a binary linear code generation matrix reduction device based on the DeepLLL reduction algorithm provided in an embodiment of the present invention. Detailed Implementation
[0040] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not limit the scope of the invention.
[0041] Existing algorithms that analogize the reduction of related lattice bases to binary linear codes still have many problems. First, the reduction conditions in lattice base reduction algorithms cannot be directly applied to the setup of binary linear codes. Furthermore, the reduction in existing LLL reduction algorithms for linear code generator matrices is essentially limited to the XOR operation between adjacent basis vectors, thus the reduction of each basis vector in the generator matrix is finite. Therefore, this invention extends the reduction conditions to compare each basis vector in the generator matrix sequentially with all subsequent basis vectors, and thus proposes a generator matrix reduction algorithm that analogizes the DeepLLL reduction algorithm of lattice bases to binary linear codes. In other words, the purpose of this invention is to solve the problem of generator matrix reduction under existing binary linear code setups, and to provide a binary linear code generator matrix reduction algorithm based on the DeepLLL reduction algorithm of lattice bases, achieving high efficiency in decoding the information set of the reduced binary linear code generator matrix. This scheme has characteristics such as low storage, high efficiency, and high decoding performance. The methods described in this invention will be explained and described in detail below:
[0042] This invention provides a method for reducing the binary linear code generation matrix based on the lattice-based DeepLLL reduction algorithm, the method comprising:
[0043] Given an arbitrary generator matrix of a binary linear code, calculate the corresponding Epipodal matrix and each Epipodal vector in the Epipodal matrix.
[0044] That is, in the embodiments of the present invention, an input is... Binary linear code Any correct generating matrix The generator matrix is generated by linearly independent Composed of dimensional vectors, denoted in sequence as follows: ,in , These are the relevant initial parameters of the linear code; "correct" means that... The epipodal vectors corresponding to each basis vector are all non-zero vectors; then the basis vectors are calculated sequentially. The corresponding Epipodal vector , where basis vectors The corresponding Epipodal vector is itself, that is... The remaining Epipodal vectors can be calculated. The calculated Epipodal vectors are used as row vectors to form a generator matrix. The corresponding Epipodal matrix .
[0045] Determine any Epipodal vector in the Epipodal matrix Is it the shortest non-zero Epipodal vector under the i-th orthogonal projection? If not, it means that there exists a basis vector. Its length under the i-th orthogonal projection is shorter. Perform Lovasz conditional judgment on the two basis vectors. If it is true, output the generator matrix and its corresponding Epipodal matrix.
[0046] Specifically, in this embodiment of the invention, the elements arranged in the original basis vectors are... After all the basis vectors are subjected to the same orthogonal projection in sequence, the original basis vectors are compared with their subsequent orthogonal projections in pairs to find the shortest non-zero Epipodal vector under each orthogonal projection, so as to make the Hamming weight distribution of each Epipodal vector more balanced.
[0047] Perform a Lovasz condition check on two basis vectors; if satisfied, then apply the Lovasz condition to the basis vectors. Perform an XOR operation, then proceed to the basis vectors that satisfy the reduction condition. Perform a basis reduction operation; otherwise, directly perform a basis reduction operation on the basis vectors that satisfy the reduction condition.
[0048] Perform basis reduction operations on the basis vectors that satisfy the reduction conditions, reducing them to the basis vectors formed by... An element in the generated basic field;
[0049] Specifically, in this embodiment of the invention, the parameters are first initialized. Then judge the second inequality. Does it hold true? If the second inequality holds true, then calculate... Conversely, calculate and in Then, continue to determine whether the second inequality holds true.
[0050] For basis vectors The insertion operation is performed, and the generated matrix and its corresponding Epipodal matrix are updated in real time. Then, it is determined whether any Epipodal vector in the Epipodal matrix is the shortest non-zero Epipodal vector under the i-th orthogonal projection.
[0051] In this embodiment of the invention, the insertion operation for the basis vector includes:
[0052] basis vectors Insert into basis vectors and Between, from the basis vectors to basis vectors All basis vectors are shifted one position to the right in sequence, i.e. ;
[0053] Update the generated matrix based on the insertion operation. and its corresponding Epipodal matrix .
[0054] In general, the method described in this embodiment of the invention selects whether the Epipodal vector corresponding to any basis vector in the given generator matrix is the shortest non-zero vector under its corresponding orthogonal projection. This is achieved by sequentially performing the same orthogonal projection on all basis vectors following the original basis vector and then comparing them pairwise. This process sequentially identifies the shortest non-zero Epipodal vector under each orthogonal projection, resulting in a more balanced Hamming weight distribution among the Epipodal vectors. Furthermore, performing a reduction operation on basis vectors that meet the relevant conditions transforms them into elements of a fundamental domain generated by a submatrix of the generator matrix, further optimizing the Hamming weights of these basis vectors. In the basis vector insertion operation, for basis vectors that have met the reduction conditions, they are simply inserted into their corresponding order positions instead of being swapped with previous basis vectors, thus improving the efficiency of the matrix update stage.
[0055] The following will combine Figure 1 The method described in the embodiments of the present invention will be explained and illustrated in detail through a specific example:
[0056] Step 1: Input the initialization parameters to obtain the specific generator matrix and its Epipodal matrix;
[0057] Step 1.1: Input a Binary linear code Any "correct" generating matrix The generator matrix is generated by linearly independent Composed of dimensional vectors, denoted in sequence as follows: ,in , These are the relevant initial parameters of the linear code; "correct" means that... The Epipodal vectors corresponding to the basis vectors are all non-zero vectors;
[0058] Step 1.2: Calculate the basis vectors sequentially. The corresponding Epipodal vector , where basis vectors The corresponding Epipodal vector is itself, that is... The remaining Epipodal vectors can be calculated. The calculated Epipodal vectors are used as row vectors to form a generator matrix. The corresponding Epipodal matrix .
[0059] Step 2: Check whether the basis vectors satisfy the reduction condition. If they do, reduce them. Otherwise, it means that all basis vectors have been reduced. That is, the generating matrix at this time is the "best" generating matrix that can be obtained under the DeepLLL reduction condition. Output the reduced generating matrix.
[0060] Step 2.1: Initialize parameters ;
[0061] Step 2.2: Calculation ;
[0062] Step 2.3: Determine the inequalities Whether it is valid, among which It refers to the first The effect under orthogonal projection, specifically the expression for its effect is: Specifically, it uses the definition of Hamming weight, which is the number of non-zero elements in a vector. This refers to the bitwise XOR operation between vectors. If the inequality is true, proceed to step 3; otherwise, proceed to steps 2 and 4.
[0063] Step 2.4: Calculation Determine the inequality Does k+1 hold true? If the inequality holds true, proceed to step 2.3; otherwise, proceed to step 2.5.
[0064] Step 2.5: Calculation Determine the inequality Does k hold true? If the inequality holds, proceed to step 2.2; otherwise, output the real-time generator matrix generated in the algorithm at this point. and its corresponding Epipodal matrix .
[0065] Step 3: Lovasz conditional judgment. This judgment is used to evaluate the basis vectors that satisfy the inequality conditions in step 2.3. Perform a preliminary reduction operation to make the corresponding Epipodal vector the shortest non-zero Epipodal vector under the i-th orthogonal projection;
[0066] Step 3.1: For the inequality in step 2.3 Two basis vectors that hold true , Perform Lovasz condition checks;
[0067] Step 3.2: Determine the inequalities Whether it is valid, among which It is a piecewise function, for ,like If it is an odd number, then ;like ,but ,in that is, vector The index of the first non-zero element; otherwise, If the inequality is true, proceed to step 3.3; otherwise, proceed to step 4.
[0068] Step 3.3: Calculation Proceed to step 4.
[0069] Step 4: Reduce the basis vectors that satisfy the reduction condition to the basis vectors. Within the fundamental domain generated by the submatrix;
[0070] Step 4.1: Initialize parameters ;
[0071] Step 4.2: Determine the inequalities Whether it is valid, among which Compared with step 3.2 The same applies. If the inequality is true, proceed to step 4.3; otherwise, proceed to step 4.4.
[0072] Step 4.3: Calculation Proceed to step 4.4;
[0073] Step 4.4: Calculation Determine the inequality Check if the inequality is true. If the inequality is true, proceed to step 4.2; otherwise, proceed to step 5.
[0074] Step 5: Perform the insertion operation. This step involves inserting the basis vectors selected in step 2.3. It is directly inserted into the position of the i-th vector in the generated matrix, because for... The selection criterion is that its Epipodal vector has a shorter length under the i-th orthogonal projection, therefore, in the case of... After performing the necessary reduction conditions in steps 3 and 4 above, it is also necessary to adjust its position in the generator matrix;
[0075] Step 5.1: Convert the basis vectors Insert into basis vectors and Between, from the basis vectors to basis vectors All basis vectors are shifted one position to the right in sequence, i.e.
[0076] ;
[0077] Step 5.2: Update the generated matrix according to the insertion operation in Step 5.1. and its corresponding Epipodal matrix Proceed to step 2.
[0078] In comparison, the method described in this embodiment of the invention involves selecting each orthogonal projection. The shortest non-zero Epipodal vector under action During the operation, same Compare them pairwise, among which That is, from the basis vectors All basis vectors from the beginning onwards could be orthogonal projections. The basis vectors corresponding to the non-zero Epipodal vectors with the shortest Hamming weights are identified. This operation can reduce or avoid duplicate alignments to some extent. Furthermore, the present invention performs further reduction operations on the basis vectors that satisfy the reduction conditions, ensuring they satisfy the fundamental domain of the submatrix generation of the corresponding generator matrix. This results in a more balanced Hamming weight distribution of the reduced Epipodal vectors.
[0079] Accordingly, embodiments of the present invention also provide a binary linear code generation matrix reduction device based on the lattice-based DeepLLL reduction algorithm, see [link to relevant documentation]. Figure 2 The device includes:
[0080] The first processing module is used to take an arbitrary generator matrix of a binary linear code as input, and calculate the Epipodal matrix corresponding to the generator matrix and each Epipodal vector in the Epipodal matrix.
[0081] The judgment module is used to determine whether any Epipodal vector in the Epipodal matrix is the shortest non-zero Epipodal vector under the corresponding i-th orthogonal projection. If not, the Lovasz condition is checked on the two basis vectors. If yes, the generated matrix and its corresponding Epipodal matrix are output. The Lovasz condition is checked on the two basis vectors. If the condition is satisfied, the basis vectors are XORed, and then the basis reduction operation is performed on the basis vectors that satisfy the reduction condition. Otherwise, the basis reduction operation is directly performed on the basis vectors that satisfy the reduction condition. The basis reduction operation is then performed on the basis vectors that satisfy the reduction condition, reducing them to an element in the generated basic field.
[0082] The second processing module is used to perform insertion operations on the basis vectors, update the generated matrix and its corresponding Epipodal matrix in real time, and then determine whether any Epipodal vector in the Epipodal matrix is the shortest non-zero Epipodal vector under the corresponding i-th orthogonal projection.
[0083] In specific implementation, the judgment module described in this embodiment of the invention is further used to perform the same orthogonal projection on all basis vectors after the original basis vector in sequence, and then compare the original basis vector with the orthogonal projections in pairs to find the shortest non-zero Epipodal vector under each orthogonal projection, so as to make the Hamming weight distribution of each Epipodal vector more balanced.
[0084] In addition, embodiments of the present invention also provide a computer-readable storage medium storing a computer program, which, when executed by a processor, implements any of the above-described binary linear code generation matrix reduction methods based on the lattice-based DeepLLL reduction algorithm.
[0085] The relevant content of the device embodiment and storage medium embodiment of the present invention can be understood by referring to the method embodiment of the present invention, and will not be discussed in detail here.
[0086] Although preferred embodiments of the invention have been disclosed for illustrative purposes, those skilled in the art will recognize that various modifications, additions, and substitutions are possible, and therefore the scope of the invention should not be limited to the embodiments described above.
Claims
1. A method for reducing the binary linear code generation matrix based on the lattice-based DeepLLL reduction algorithm, characterized in that, include: Given an arbitrary generator matrix of a binary linear code, calculate the corresponding Epipodal matrix and each Epipodal vector in the Epipodal matrix. Determine any Epipodal vector in the Epipodal matrix Is it the shortest non-zero Epipodal vector under the i-th orthogonal projection? If not, it means that a basis vector exists. The length of the vector under the i-th orthogonal projection is shorter. Lovasz conditional judgment is performed on the two basis vectors. If the condition is met, the generator matrix and its corresponding Epipodal matrix are output. Perform a Lovasz condition check on two basis vectors; if satisfied, then apply the Lovasz condition to the basis vectors. Perform an XOR operation, then proceed to the basis vectors that satisfy the reduction condition. Perform a basis reduction operation; otherwise, directly perform basis reduction on the basis vectors that satisfy the reduction condition, reducing them to a subset of the vectors defined by the basis reduction condition. An element in the generated basic field; For basis vectors The insertion operation is performed, and the generated matrix and its corresponding Epipodal matrix are updated in real time. Then, it is determined whether any Epipodal vector in the Epipodal matrix is the shortest non-zero Epipodal vector under the i-th orthogonal projection.
2. The method according to claim 1, characterized in that, The input is an arbitrary generator matrix of a binary linear code, and the calculation of the corresponding Epipodal matrix and each Epipodal vector in the Epipodal matrix includes: Enter one Binary linear code Any "correct" generating matrix The generator matrix is generated by linearly independent Composed of dimensional vectors, denoted in sequence as follows: ,in , These are the relevant initial parameters of the linear code; "correct" means that... The Epipodal vectors corresponding to the basis vectors are all non-zero vectors; Calculate the basis vectors sequentially The corresponding Epipodal vector , where basis vectors The corresponding Epipodal vector is itself, that is... The remaining Epipodal vectors can be calculated based on... The calculated Epipodal vectors are used as row vectors to form a generator matrix. The corresponding Epipodal matrix .
3. The method according to claim 1, characterized in that, Determine any Epipodal vector in the Epipodal matrix Is it the shortest non-zero Epipodal vector corresponding to the i-th orthogonal projection, including: The original basis vectors will be ranked After all the basis vectors are subjected to the same orthogonal projection in sequence, the original basis vectors are compared with their subsequent orthogonal projections in pairs to find the shortest non-zero Epipodal vector under each orthogonal projection, so as to make the Hamming weight distribution of each Epipodal vector more balanced.
4. The method according to claim 3, characterized in that, Determining whether any Epipodal vector in the Epipodal matrix is the shortest non-zero Epipodal vector under the corresponding i-th orthogonal projection also includes: Initialization parameters ,calculate ; Determine the first inequality Whether it is valid, among which It refers to the first The effect under orthographic projection This refers to the bitwise XOR operation between vectors. If the first inequality shown is true, then the Lovasz condition is checked on the two basis vectors; otherwise, the calculation is performed. And further judgment If k+1 holds true, then further determine whether the first inequality holds true. Is it true? Otherwise, calculate. ,judge If k holds true, then calculate... Conversely, output the real-time generator matrix in the algorithm at this point. and its corresponding Epipodal matrix ; in, .
5. The method according to claim 4, characterized in that, The Lovasz conditional judgment on the two basis vectors includes: For those that make the first inequality Two basis vectors that hold true , Perform Lovasz condition checks; Determine the second inequality Whether it is valid, among which It is a piecewise function, for ,like If it is an odd number, then ;like ,but ,in that is, vector The index of the first non-zero element; otherwise, If the second inequality holds, then calculate .
6. The method according to claim 5, characterized in that, The basis vectors satisfying the reduction condition are reduced to basis reduction operations, reducing them to the form given by... An element in the generated basic field includes: Initialization parameters ; Determine the second inequality Does it hold true? If the second inequality holds true, then calculate... Conversely, calculate and in Then, continue to determine whether the second inequality holds true.
7. The method according to claim 1, characterized in that, The insertion operation for the basis vectors includes: basis vectors Insert into basis vectors and Between, from the basis vectors to basis vectors All basis vectors are shifted one position to the right in sequence, i.e. ; Update the generated matrix based on the insertion operation. and its corresponding Epipodal matrix .
8. A binary linear code generation matrix reduction device based on the lattice-based DeepLLL reduction algorithm, characterized in that, The device includes: The first processing module is used to take an arbitrary generator matrix of a binary linear code as input, and calculate the Epipodal matrix corresponding to the generator matrix and each Epipodal vector in the Epipodal matrix. The judgment module is used to determine whether any Epipodal vector in the Epipodal matrix is the shortest non-zero Epipodal vector under the corresponding i-th orthogonal projection. If not, the Lovasz condition is checked on the two basis vectors. If yes, the generated matrix and its corresponding Epipodal matrix are output. The Lovasz condition is checked on the two basis vectors. If the condition is satisfied, the basis vectors are XORed, and then the basis reduction operation is performed on the basis vectors that satisfy the reduction condition. Otherwise, the basis reduction operation is directly performed on the basis vectors that satisfy the reduction condition. The basis reduction operation is then performed on the basis vectors that satisfy the reduction condition, reducing them to an element in the generated basic field. The second processing module is used to perform insertion operations on the basis vectors, update the generated matrix and its corresponding Epipodal matrix in real time, and then determine whether any Epipodal vector in the Epipodal matrix is the shortest non-zero Epipodal vector under the corresponding i-th orthogonal projection.
9. The apparatus according to claim 8, characterized in that, The judgment module is further configured to perform the same orthogonal projection on all basis vectors following the original basis vector in sequence, and then compare the original basis vector with the subsequent orthogonal projections in pairs to find the shortest non-zero Epipodal vector under each orthogonal projection, so as to make the Hamming weight distribution of each Epipodal vector more balanced.
10. A computer-readable storage medium storing a computer program that, when executed by a processor, implements the binary linear code generation matrix reduction method based on the lattice-based DeepLLL reduction algorithm as described in any one of claims 1-7.