CN111740747B - A construction method of low-rank circulant matrix and its associated multivariate LDPC code - Google Patents

Abstract

The application relates to the technical field of wireless communication technology and image processing, and discloses a construction method of a low-rank cyclic matrix and a related multi-element LDPC code thereof, wherein the search space of the cyclic matrix is reduced by utilizing isomorphic theory; and searching by using a rank calculation algorithm to obtain cyclic matrixes with different ranks. The application obtains the low-rank cyclic matrix which is easy to realize by hardware, and provides a construction method of the low-rank cyclic matrix. A non-zero field element assignment method is adopted, and a multi-element LDPC code construction method under different orders is provided. Compared with binary LDPC codes constructed based on PEG algorithm, the constructed multi-element LDPC codes have the error code rate of 10 under the BPSK modulation mode ^‑5 A coding gain of 0.9dB in the vicinity; when combined with higher order modulation, there is a greater performance improvement. In the additive white gaussian noise channel, the constructed multi-element LDPC code has good iterative decoding performance, and the performance curves under the iteration of 5 times and 50 times almost overlap.

Description

A construction method of low-rank circulant matrix and its associated multivariate LDPC code

Technical field

The present invention relates to the fields of wireless communication technology and image processing technology, and in particular to a construction method of a low-rank cyclic matrix and its associated multivariate LDPC code.

Background technique

At present, the basic function stage of 5G standardization has been completed, and the next stage of standardization is mainly aimed at the main Internet of Things/vertical industry application scenarios, providing wireless communication transmission solutions that support the information society in the next 10 years. Standardization mainly includes two aspects: high-reliability low-latency communication service (URLLC) and large-scale machine communication (mMTC). Different from 5G, 6G's vision of "everything goes where you want" needs to meet the requirements of real-time, reliability, throughput and massive connections at the same time, which will pose new challenges to the new generation of wireless communication networks. Delay and reliability indicators are usually considered together and refer to the maximum transmission delay of the communication system under a certain probability of correct transmission. For channel coding, it is required to have low coding and decoding processing delays and to eliminate the error levels generated by the decoding algorithm. Combined with soft output iterative decoding, LDPC codes are a competitive and practical channel coding technology. Research shows that under short and medium code lengths, compared with binary LDPC codes under the same bit length, multivariate LDPC codes have the following advantages: 1) more (1~1.3dB) coding gain; 2) stronger Anti-burst error capability; 3) Easier to combine with high-order modulation. In recent years, under the iterative decoding framework, the problem of high decoding complexity of multivariate LDPC codes has been effectively solved, which laid a solid foundation for the application of multivariate LDPC codes. In iterative decoding, the redundant rows of the LDPC code check matrix can speed up the decoding convergence speed, thereby effectively reducing the decoding delay. In addition, in image processing, the data matrices of natural images are usually low-rank or approximately low-rank. That is to say, each row (or column) of these matrices can be linearly represented by other rows (or columns), thus containing a large amount of redundant information. Based on this redundant information, the noise information of the image can be removed, and the true image information can be restored, and the erroneous image information can also be restored. However, there are relatively few studies on low-rank matrix construction. In summary, it is very meaningful to study the construction method of low-rank matrices (or matrices with more redundant rows).

Circular matrices have circular shift properties and are easily implemented in hardware based on linear shift registers. At present, the number of cyclic matrices constructed based on Euclidean geometry, Reed-Solomon codes, and two-dimensional maximum distance separable codes is limited; and the computer exhaustive search method based on cyclic codes and isomorphism theory, with the size of the matrix and row (or As the column weight increases, the search space will increase sharply, and it will become extremely difficult to find and determine different structural classes.

Through the above analysis, the problems and defects existing in the existing technology are: the number of cyclic matrices constructed based on algebraic structures is extremely limited, which is difficult to meet the needs of practical applications; and the computational complexity of computer search algorithms based on cyclic codes and isomorphism theory is very high High, the search space is very large, especially when the matrix and row (column) weight are large, it is extremely difficult to complete the exhaustive search.

The difficulty of solving the above problems and defects is: it is very difficult to find new algebraic structures to construct circulant matrices; when the matrix and row (column) weight are large, the computer-based search space is very large, and it is difficult to complete the exhaustive search solution, and Finding non-isomorphic matrices is also unrealistic.

The significance of solving the above problems and defects is: the present invention converts the search space of the matrix into the search space of the position set, and finds circulant matrices of different ranks based on the rank algorithm. This provides a variety of low-rank circulant matrices for image processing technology. In addition, the present invention also proposes a construction method of multivariate LDPC codes based on the assignment method of multivariate domain elements. This provides a candidate coding scheme for high-reliability and low-latency communication services in the new generation of mobile communication systems.

Contents of the invention

In view of the problems existing in the prior art, the present invention provides a construction method of a low-rank circulant matrix and its associated multivariate LDPC code.

The present invention is implemented as follows: a construction method of a low-rank circulant matrix. The construction method of a low-rank circulant matrix includes: using isomorphism theory to reduce the search space of the circulant matrix; using a rank algorithm to search to obtain circulant matrices of different ranks. . A method of constructing a multivariate LDPC code. The construction method of the LDPC code includes: obtaining multivariate LDPC codes of different orders and different code rates based on a multivariate domain assignment method and a low-rank circulant matrix.

Further, the construction method of the low-rank circulant matrix is to let C ₁ and C ₂ be two binary circulant matrices with row or column weight m and size L×L, and their first row non-zero position sets are respectively recorded as S ₁ ={s _1,1 ,s _1,2 ,s _1,3 ,...,s _1,m } and S ₂ ={s _2,1 ,s _2,2 ,s _2,3 ,... .,s _2,m }, if the circulant matrix C ₂ is obtained from C ₁ according to at least one of the following conditions, then C ₁ is said to be isomorphic to C ₂ , recorded as

For constant c∈{0,1,2,...,L-1}, all elements of set S ₂ can be obtained by adding all elements of set S ₁ plus a constant c, that is, for 1≤i≤m, s _2,i =s _1,i +c(mod L), c∈{1,2,...,L-1}, and is relatively prime with L;

All elements of set S ₂ and all elements of set S ₁ satisfy the following equation: for 1≤i≤m,s _2,i =c·s _1,i (mod L).

Further, the construction method of the low-rank circulant matrix includes: given the row number L and the row or column weight m of the circulant matrix C, constructing the circulant matrix C is equivalent to designing the non-zero element position set S={s of the first row ₁ , s ₂ , s ₃ ,..., s _m }, that is, a set whose cardinality is m, construct a position set S={s ₁ , s ₂ , s ₃ ,... ,s _m }, where, for 1≤i＜j≤m, 0≤s _i ＜s _j ≤L-1;

It can be seen from the number and value range of elements in the set S that the total number of position sets S is:

Any position set S is isomorphic to a position set S _- containing 0 elements:

The subtraction operation in the set S _- is performed modulo L, and the element s ₁ in the position set S is directly set to 0. It can be seen from the non-repeatability of the position set that the total number of the position set S is reduced to:

The elements (s ₂ -s ₁ ) in the set S _- are relatively prime with L. From the knowledge of number theory, it can be known that there is a number n such that (s ₂ -s ₁ )·n=1 (mod L), then, the set S _-Isomorphic to a position set S _* containing 0 elements and 1 elements, that is:

The multiplication operation in the set S _* is performed modulo L. The element s ₁ in the position set S is directly set to 0 and the element s ₂ is set to 1. It can be seen from the non-repeatability of the position set that the total number of the position set S The number is reduced to:

The minimum value of the circulant matrix rank is 1 and the maximum value is L. Set a threshold value R and only need to find a circulant matrix with a rank less than R.

Further, the search method for the low-rank circulant matrix includes:

Step 1: Select (m-1) elements from the set {1,2,...,L-1}, and select a set S of non-zero element positions in the order of combination;

Step 2: Generate a circulant matrix C based on the position set S in step 1;

Step 3: Calculate the rank r of the circulant matrix C in Step 2;

Step 4: If r is less than R, store the location set S and record its rank as r;

Step 5: Repeat steps 1 to 4 until all position sets with ranks from 1 to R are found, or all are found. set of locations.

Another object of the present invention is to provide a multivariate LDPC code construction method based on the construction method of the low-rank circulant matrix. The multivariate LDPC code construction method includes: a search method based on the low-rank circulant matrix and a check method in the circulant matrix C Is there a 4-ring algorithm to obtain a binary circulant matrix C of size L×L, whose Tanner diagram has a girth of at least 6? Replace the non-zero element 1 in the circulant matrix C with the finite field GF(q) During the replacement process, the obtained multivariate matrix is not of full rank. Directly replace the non-zero element 1 in the matrix C randomly with non-zero field elements, then the obtained multivariate matrix is basically Full rank; using the non-zero domain element assignment method, based on a binary circulant matrix, a set of multivariate LDPC codes with flexible domain order and code rate are obtained. The rank of the binary circulant matrix C is R, and the multivariate LDPC code The selectable code rates are 1-R, 1-R-1/L, 1-R-2/L, 1-R-3/L,...,0.

Further, the algorithm for checking whether there is a 4-ring in the circulant matrix C by the multivariate LDPC code construction method includes:

Step 1: Obtain the difference set D={d∈Z _L |d=s _i -s _j (modL),1≤i based on the position set S={s ₁ , s ₂ , s ₃ ,..., s _m } ≤m,1≤j≤m,i≠j};

Step 2: Calculate the number of elements in the difference set D, or check whether there is in the set E={e|e=d _i +d _j (modL),i≠j,d _i ∈D,d _j ∈D,} zero element;

Step 3: If the number of elements in the difference set D is less than Or there are zero elements in the set E, and the direct output is that there is a 4-ring; otherwise, the output is that there is no 4-ring.

Further, the assignment method of non-zero domain elements in the multivariate LDPC code construction method includes:

Method 1: Replace all non-zero elements 1 in each column of the binary circulant matrix C with the same non-zero field element on the finite field GF(q). The non-zero field elements here are randomly selected to obtain a GF( The matrix C _q on q), the null space of the matrix C _q gives a set of q-element LDPC codes with code rate (1-R) and code length L;

Method 2: Replace the non-zero element 1 of a certain column or columns in the binary circulant matrix C with a different non-zero field element on the finite field GF(q), and replace the non-zero element 1 of each remaining column with a finite The same non-zero field element on the field GF(q), the non-zero field element is randomly selected, and a matrix C _q on the GF(q) is obtained. Generally, as the number of columns with different non-zero domain elements in the matrix C _q gradually increases, the rank of the matrix C _q will increase one by one until it is full rank. The null space of the matrix C _q defines a set of q elements with variable code rates. LDPC code.

Another object of the present invention is to provide an image processing technology. The image processing technology includes matrix restoration and matrix decomposition technology. The matrix restoration and matrix decomposition require low-rank matrices. The construction method of the low-rank circulant matrix is executed. The steps are as follows: use isomorphism theory to reduce the search space of the circulant matrix; use the rank algorithm to search to obtain circulant matrices of different ranks.

Another object of the present invention is to provide a channel coding scheme. The channel coding scheme includes a multivariate LDPC code. The construction method of the multivariate LDPC code performs the following steps: using the assignment method of the multivariate domain and the low-rank cycle The matrix construction method can obtain multivariate LDPC codes of different orders and different code rates.

Combined with all the above technical solutions, the advantages and positive effects of the present invention are: in image processing, the redundant information of the low-rank matrix can be used for image restoration and image feature extraction, while in iterative decoding, the check matrix Redundant rows can speed up decoding convergence. The present invention obtains a low-rank circulant matrix that is easy to implement in hardware. The isomorphism theory of circulant matrices is discussed, and based on this, a construction method of low-rank circulant matrices is proposed. Considering the impact of the length 4 ring in the Tanner graph on iterative decoding performance, a multivariate LDPC code construction method under different orders is proposed using the assignment method of non-zero domain elements. Numerical simulation results show that compared with the binary LDPC code constructed based on the PEG algorithm, the constructed multivariate LDPC code has a coding gain of 0.9dB near the bit error rate of 10 ^-5 in the BPSK modulation mode; compared with high-order modulation, When combined, there is greater performance improvement. In addition, the performance of the constructed multivariate LDPC code under 5 and 50 iterations is almost the same, which provides an effective candidate coding scheme for low-latency and high-reliability communication.

The present invention adopts channel coding technology for low-latency and high-reliability communication. These communication services are mainly oriented to the Internet of Things represented by machine communication. They have the characteristics of small data packets, low power consumption, massive connections, strong burstiness, etc., and require encoding and decoding. Channel coding scheme with fast speed, strong burst resistance and short code length. In the construction method of the low-rank circulant matrix of the present invention, the circulant matrix refers to a square matrix with a size of Left) circular shift; each of its columns is a downward (or upward) circular shift of the column to its left, and the first column is a downward (or upward) circular shift of the last column. Obviously, the row weight and column weight of a circulant matrix are the same.

The present invention first uses isomorphism theory to reduce the search space of the circulant matrix, and uses the rank algorithm to search to obtain circulant matrices of different ranks. The method of calculating ranks is used to directly find circulant matrices of different ranks, instead of searching and dividing isomorphic classes of circulant matrices. Furthermore, through the ring structure of length 4 (abbreviated as 4-ring) in the circulant matrix Tanner diagram, an algorithm for determining the 4-ring is proposed. A non-zero element assignment method is also given, and a girth of at least 6 is proposed. Multivariate LDPC code construction method. Numerical simulation results show that in the Additive White Gaussian Noise (AWGN) channel, the constructed multivariate LDPC code has good iterative decoding performance, and the performance curves under 5 and 50 iterations are almost overlapping.

Description of the drawings

In order to explain the technical solutions of the embodiments of the present application more clearly, the following will briefly introduce the drawings required to be used in the embodiments of the present application. Obviously, the drawings described below are only some embodiments of the present application. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without exerting creative efforts.

Figure 1 is a flow chart of a method for constructing a low-rank circulant matrix provided by an embodiment of the present invention.

Figure 2 is a schematic diagram comparing the bit error rate performance of the (31,15) LDPC code on GF (64) and the binary (186,90) LDPC code constructed based on the PEG algorithm under different iteration numbers provided by the embodiment of the present invention. .

Figure 3 is a schematic diagram comparing the bit error rate performance of the (31,15) LDPC code on GF (64) and the binary (186,90) LDPC code constructed based on the PEG algorithm under high-order modulation provided by the embodiment of the present invention. .

Figure 4 is the bit error rate of (31, 15) LDPC codes on GF (4), GF (8), GF (32) and GF (128) provided by the embodiment of the present invention under 5 and 50 iterations. Performance diagram.

Detailed ways

In order to make the purpose, technical solutions and advantages of the present invention more clear, the present invention will be further described in detail below in conjunction with the examples. It should be understood that the specific embodiments described here are only used to explain the present invention and are not intended to limit the present invention.

In view of the problems existing in the prior art, the present invention provides a construction method of a low-rank circulant matrix and its associated multivariate LDPC code. The present invention will be described in detail below with reference to the accompanying drawings.

As shown in Figure 1, the construction method of low-rank circulant matrix and multivariate LDPC code provided by the present invention includes the following steps:

S101: Use isomorphism theory to reduce the search space of circulant matrices;

S102: Use the rank algorithm to search to obtain circulant matrices of different ranks;

S103: Use the 4-ring calculation algorithm to obtain the circulant matrix without 4-rings;

S104: Use the domain element assignment method and the circulant matrix without 4-rings to obtain multivariate LDPC codes of different orders and different code rates.

The present invention constructs a low-rank circulant matrix and a multivariate LDPC code to give the following embodiment:

Embodiment 1: Construct a circulant matrix with a row (or column) number of 31 and a row (or column) weight of 5 and a set of code lengths of 31 and code rates of 15/31, 14/31, 13/31, 12/ 31, 11/31, 10/31, 9/31, 8/31, 7/31, 6/31, 5/31, 4/31, 3/31, 2/31, 1/31 multiplex LDPC codes.

Referring to Figure 1, the implementation steps of the present invention are as follows:

Step 1. According to the circulant matrix to be constructed with a row (or column) number of 31 and a row (or column) weight of 5, you might as well make its corresponding position set as S={s ₁ , s ₂ , s ₃ , s ₄ ,s ₅ }, where, for 1≤i＜j≤5, 0≤s _i ＜s _j≤30 . Obviously, the search space of the location set S is Based on the isomorphism theory, the position set can be simplified as S ^* ={0,s ₁ ,s ₂ ,s ₃ ,s ₄ }, where, for 1≤i＜j≤4,1≤s _i ＜s _j≤30 . Obviously, the search space of the location set S ^* is/>

Step 2: According to the search space of the position set S ^* in step 1, use the rank algorithm to obtain three circulant matrices of different ranks. Their corresponding ranks are 16 and the position sets are {0, 1, 3, 7, 15} , the rank is 21, the position set is {0, 1, 2, 6, 27}, the rank is 26, the position set is {0, 1, 2, 3, 5}.

Step 3. Based on the three circulant matrices obtained in step 2, using the 4-ring calculation algorithm, two circulant matrices without 4-rings can be obtained. Their corresponding rank is 16 and the position set is {0, 1, 3. , 7, 15} and the rank is 26, and the position set is {0, 1, 2, 3, 5}.

Step 4. According to the circulant matrix with rank 16 and position set {0, 1, 3, 7, 15} obtained in step 3, using the domain element assignment method, a set of code length 31 and variable code rate can be obtained The q-yuan LDPC code has selectable code rates: {15/31, 14/31, 13/31, 12/31, 11/31, 10/31, 9/31, 8/31, 7/31, 6/31, 5/31, 4/31, 3/31, 2/31, 1/31}.

The low-rank circulant matrix construction method provided by the present invention can also be implemented by those of ordinary skill in the industry using other steps. The low-rank circulant matrix construction method provided by the present invention in Figure 1 is only a specific embodiment.

The technical solution of the present invention will be further described below with reference to the accompanying drawings.

1. Low-rank circulant matrix construction method based on isomorphism theory

1.1 Circular matrix and its isomorphism theory

Here, the circulant matrix C=[c _i,j ] _L×L considered in the present invention is recorded as a binary matrix with a row (or column) weight of m and a size of L×L. Due to the cyclic shift characteristics of the circulant matrix C, the present invention only needs to mark the non-zero element position of the first row of the circulant matrix C. Let us assume that the set of non-zero element positions is S={s ₁ , s ₂ , s ₃ ,..., s _m }, where, for 1≤i＜j≤m, 0≤s _i ＜s _j ≤L-1 . Therefore, the construction of the circulant matrix C is equivalent to the design of the first row non-zero element position set S.

The Tanner graph of the circulant matrix C is a bipartite graph. The nodes in the Tanner diagram are divided into two categories: Variable Node (or encoding bit node) and Check Node (or constraint node), represented by VN and CN respectively. Lines in Tanner diagrams only connect nodes of different types. The Tanner graph of the circulant matrix C can be obtained as follows: when the element c _i,j in C is 1, the i-th check node (CN i) and the j-th variable node (VNj) are connected; otherwise there is no connection between them lines connected. The length of the shortest ring in the Tanner diagram of the circulant matrix C is called the girth. If the Tanner graphs of two circulant matrices are isomorphic, then the two circulant matrices are also said to be isomorphic. According to Theorem 2 in the literature [XU Hengzhou, BAI Baoming, ZHU Min, et al. Construction of short-blocknonbinary LDPC codes based on cyclic codes [J]. China Communications, 2017, 14(8):1-9], the following Give the isomorphism theorem of circulant matrices without proof.

Theorem 1 (isomorphism theory of circulant matrices): Let C ₁ and C ₂ be two binary circulant matrices with row (or column) weight m and size L × L. Their first row non-zero position sets are respectively Denoted as S ₁ ={s _1,1 ,s _1,2 ,s _1,3 ,...,s _1,m } and S ₂ ={s _2,1 ,s _2,2 ,s _2,3 , ...,s _2,m }. If the circulant matrix C ₂ can be obtained from C ₁ according to at least one of the following conditions, then C ₁ is said to be isomorphic to C ₂ , denoted as

For constant c∈{0,1,2,...,L-1}, all elements of set S ₂ can be obtained by adding all elements of set S ₁ plus a constant c, that is, for 1≤i≤m, s _2,i =s _1,i +c(mod L). Assume that c∈{1,2,...,L-1}, and is relatively prime with L.

All elements of set S ₂ and all elements of set S ₁ satisfy the following equation: for 1≤i≤m,s _2,i =c·s _1,i (mod L).

1.2 Low-rank circulant matrix construction method based on isomorphism theory

Given the number of rows L and the row (or column) weight m of the circulant matrix C, constructing the circulant matrix C is equivalent to designing the set of non-zero element positions of the first row S = {s ₁ , s ₂ , s ₃ ,... ,s _m }, that is, a set whose basis (Cardinality) is m. Therefore, the present invention mainly constructs a position set S={s ₁ , s ₂ , s ₃ ,..., s _m } with a basis of m, where, for 1≤i<j≤m, 0≤s _i <s _j≤L -1.

It can be seen from the number and value range of elements in the set S that the total number of position sets S is:

It can be seen from the first condition of Theorem 1 that any position set S is isomorphic to a position set S _- containing 0 elements, that is:

Note that the subtraction operation in the set S _- is performed modulo L. Therefore, the element s ₁ in the position set S can be directly set to 0. From the non-repeatability of the position set, the total number of the position set S is reduced to:

This effectively reduces the search space of the location set S. Assume that the elements (s ₂ -s ₁ ) in the set S _- are relatively prime to L. From the knowledge of number theory, it can be known that there is a number n such that (s ₂ -s ₁ )·n=1 (mod L). Then, from the second condition of Theorem 1, it can be seen that the set S _- is isomorphic to a position set S _* containing 0 elements and 1 elements, that is:

Note that the multiplication operations in the set S _* are performed modulo L. In this case, you can directly set the element s ₁ in the position set S to 0 and the element s ₂ to 1. From the non-repeatability of the position set, the total number of the position set S is reduced to:

This can further reduce the search space of the location set S. Due to practical requirements, the present invention only needs to construct a circulant matrix with a specific rank. It can be seen from the size of the circulant matrix that the minimum value of the circulant matrix rank is 1 and the maximum value is L. Since the present invention mainly focuses on low-rank matrices, in order to reduce the search space, a threshold R is set here, and only circulant matrices with a rank less than R are needed. A search algorithm for constructing a low-rank circulant matrix is given below, namely Algorithm 1.

Algorithm 1 Circular matrix search algorithm with rank less than R

In order to prove the effectiveness of Algorithm 1, Table 1 gives some search results of low-rank circulant matrices.

Table 1 Partial circulant matrix searched based on Algorithm 1

2. Multivariate LDPC code construction based on low-rank circulant matrix

2.1 4-ring structure of circulant matrix

Short rings, especially 4-rings, will reduce the iterative decoding performance of LDPC codes. Therefore, the 4-ring structure of the circulant matrix is analyzed, and a method for determining the 4-ring of the circulant matrix is given.

4-ring structure in circulant matrix C:

The 4-ring in the circulant matrix C consists of four 1 elements, which are distributed in two rows and two columns. Its structure is shown in Figure 1. From the relationship between the circulant matrix and the position set S={s ₁ , s ₂ , s ₃ ,..., s _m }, it can be seen that the row and column coordinates of these four 1 elements can be abbreviated as:

(a _, s _i +a), (b, s _k +b), (b, s _l +b), (a, s _j +a);

Among them, 0≤a＜b≤L-1, 1≤i＜j≤m, 1≤k＜l≤m,. Note that the addition operation in the parentheses of the above formula is based on modulo L. Obviously, the necessary and sufficient condition for the existence of this 4-ring in Figure 2 is that one of the following formulas holds:

s _i -s _j =s _k -s _l (mod L) (1)

or

(s _i -s _j )+(s _l -s _k )=0(mod L) (2)

Since i≠j,k≠l,(i,j)≠(k,l), whether the above two formulas are true depends on two numbers, namely (s _i -s _j ) and (s _l -s _k ). Note that these two numbers are positive numbers obtained modulo L. Based on this, a new concept "difference set" is defined here based on the location set S.

Definition 1 (difference set of position sets): Let the position set of circulant matrix C be S={s ₁ , s ₂ , s ₃ ,..., s _m }, and the number of rows (or columns) of circulant matrix C is L. The difference set of the position set S is D={d∈Z _L |d=s _i -s _j (mod L), 1≤i≤m, 1≤j≤m, i≠j}.

Obviously, the difference set is also a set. Theoretically, the basis of this set is Due to the non-repeatability of sets, if the number of elements of the difference set D is less than/> This means that at least two elements in the difference set are equal, which also means that formula (1) is true. Or, when the sum of the two elements in the difference set D equals zero modulo L, then equation (2) is established. This also corresponds to the existence of a 4-ring. This shows that there is at least one 4-ring in the circulant matrix C. Based on this, the present invention provides an algorithm for checking whether a 4-ring exists in the circulant matrix C, namely Algorithm 2.

Algorithm 2 is an algorithm for checking whether there is a 4-ring in the circulant matrix C.

According to Algorithm 1 and Algorithm 2, the present invention can obtain some low-rank circulant matrix position sets that do not contain 4-rings. In order to prove the effectiveness of Algorithm 2, Table 2 gives some position sets whose corresponding circulant matrix Tanner graph does not have 4-rings.

Table 2 The set of circulant matrix positions that does not contain 4-rings

2.2 Construction method of multivariate LDPC code

The present invention mainly studies the construction method of multivariate LDPC codes. Based on Algorithm 1 and Algorithm 2, a binary circulant matrix C of size L×L can be obtained, and the girth of its Tanner diagram is at least 6. In order to construct a multivariate matrix, it is also necessary to replace the non-zero element 1 in the circulant matrix C with a non-zero field element on the finite field GF(q). It is worth noting that during the replacement process, it is also necessary to ensure that the obtained multivariate matrix is not full rank. Normally, by directly replacing the non-zero elements 1 in the matrix C with non-zero field elements, the resulting multivariate matrix will basically be of full rank. Therefore, the present invention adopts the non-zero domain element assignment method and is based on a binary circulant matrix to obtain a set of multivariate LDPC codes with flexible domain order and code rate. Assume that the rank of the binary circulant matrix C is R. Then, the optional code rate of the multivariate LDPC code proposed by the present invention is 1-R, 1-R-1/L, 1-R-2/L, 1- R-3/L,...,0. The following is a brief introduction to two methods of assigning values to non-zero field elements.

Method 1: Replace all non-zero elements 1 in each column of the binary circulant matrix C with the same non-zero field element on the finite field GF(q), where the non-zero field elements are randomly selected. In this way, a matrix C _q on GF(q) can be obtained. According to Theorem 1 of the literature [XU Hengzhou, FENG Dan, SUN Cheng, et al.Algebraic-based nonbinary LDPCcodes with flexible field orders and code rates[J].China Communications, 2017, 14(4):111-119], it can be known that, The binary circulant matrix C has the same rank as the multivariate matrix C _q . Therefore, the null space of matrix C _q gives a set of q-element LDPC codes with code rate (1-R) and code length L.

Method 2: Replace the non-zero element 1 of a certain column (or columns) in the binary circulant matrix C with a different non-zero field element (requirements are different) on the finite field GF(q), and the remaining columns of The non-zero element 1 is replaced by the same non-zero field element on the finite field GF(q), where the non-zero field element is randomly selected. In this way, a matrix C _q on GF(q) can be obtained. Generally, as the number of columns with different non-zero domain elements in the columns of matrix C _q gradually increases, the rank of matrix C _q will increase one by one until it reaches full rank. Therefore, the null space of matrix C _q can define a set of q-element LDPC codes with variable code rate.

The technical effects of the present invention will be described in detail below in conjunction with simulation.

1. Simulation results

The simulation parameters below are for the AWGN channel and BPSK modulation. The decoding algorithm of binary LDPC codes is sum-product algorithm (SPA), while the decoding algorithm of multivariate LDPC codes is multivariate sum-product algorithm (QSPA) based on fast Fourier transform (FFT). The selected high-order modulations are QPSK, 8PSK modulation and 64-QAM.

Consider a circulant matrix with a row (or column) number of 31 and a row (or column) weight of 5. According to Table 2, a circulant matrix without 4-rings can be found, its position matrix is {0, 1, 3, 7, 15} and its rank is 16. According to method 1, a set of q-element LDPC codes with a code length of 31 and a code rate of 15/31 can be constructed. According to method 2, a set of q-element LDPC codes with a code length of 31 and a variable code rate can be obtained. The optional code rates are {15/31, 14/31, 13/31, 12/31, 11/31 , 10/31, 9/31, 8/31, 7/31, 6/31, 5/31, 4/31, 3/31, 2/31, 1/31}. According to method 1, by selecting finite field GF (64), a 64-element (31, 15) LDPC code can be obtained. Figure 3 shows the word error rate (WER) performance of the code using QSPA with 1, 3, 5 and 50 iterations. In order to compare under the same code parameters (equivalent bit code length and code rate), a binary (186, 90) LDPC code is constructed based on the PEG algorithm. Figure 3 also shows the bit error rate performance of the code using SPA with 5 and 50 iterations and the finite length performance limit of a code length of 186 bits and a code rate of 15/31. It can be seen that when the number of iterations is 50 and the bit error rate is equal to 10 ^-5 , the constructed 64-element (31, 15) LDPC code has a coding gain of approximately 0.9dB compared to the binary (186, 90) LDPC code. When the number of iterations is 5, the performance gap is about 1.8dB. In addition, it can also be seen that the performance gap between the constructed 64-element (31,15) LDPC code between 5 and 50 iterations is very small; when the bit error rate is equal to 10 ^-5 , the constructed 64-element (31 , 15) The LDPC code is about 1dB away from the finite length performance limit. Figure 4 shows the bit error rate performance of the constructed 64-element (31, 15) LDPC code and binary (186, 90) LDPC code under high-order modulation. It can be seen that as the modulation order increases, the performance gap between the constructed multivariate code and the binary code also becomes larger, and the performance curves of the constructed multivariate code at 5 and 50 iterations almost overlap. According to method 1, four (31,15) multivariate LDPC codes can be obtained by selecting finite fields GF(4), GF(8), GF(32) and GF(128). Figure 4 shows the bit error rate performance of these four codes under QSPA with 5 and 50 iterations. It can be seen from Figure 4 that the constructed multivariate LDPC code has better decoding performance, and there is no error level at the bit error rate of 10 ^-6 . In addition, the proposed multivariate LDPC code only needs 5 iterations to achieve the decoding performance of 50 iterations.

The present invention adopts a construction method of a type of low-rank circulant matrix. First, the construction of a circulant matrix is transformed into the design of a non-zero element position set, and a search algorithm for a low-rank circulant matrix is proposed based on the isomorphism theory of position sets. Furthermore, the 4-ring structure of the circulant matrix was analyzed, and a circulant matrix with a girth of at least 6 was obtained. Based on this, using two assignment methods of non-zero domain elements, a construction method of multivariate LDPC codes is proposed. Numerical simulation results on the AWGN channel show that the constructed multivariate LDPC code has better decoding performance, and only needs 5 iterations to achieve the decoding performance of 50 iterations. This provides an effective candidate coding scheme for low-latency and highly reliable wireless communications. In order to further improve the performance of such codes, it is worth studying how to optimize their non-zero domain elements.

It should be noted that embodiments of the present invention may be implemented by hardware, software, or a combination of software and hardware. The hardware part can be implemented using dedicated logic; the software part can be stored in memory and executed by an appropriate instruction execution system, such as a microprocessor or specially designed hardware. Those of ordinary skill in the art will understand that the above-described apparatus and methods may be implemented using computer-executable instructions and/or included in processor control code, for example on a carrier medium such as a disk, CD or DVD-ROM, such as a read-only memory. Such code is provided on a programmable memory (firmware) or on a data carrier such as an optical or electronic signal carrier. The device and its modules of the present invention may be implemented by hardware circuits such as very large scale integrated circuits or gate arrays, semiconductors such as logic chips, transistors, etc., or programmable hardware devices such as field programmable gate arrays, programmable logic devices, etc., It can also be implemented by software executed by various types of processors, or by a combination of the above-mentioned hardware circuits and software, such as firmware.

The above are only specific embodiments of the present invention, but the protection scope of the present invention is not limited thereto. Any person familiar with the technical field shall, within the technical scope disclosed in the present invention, be within the spirit and principles of the present invention. Any modifications, equivalent substitutions and improvements made within the above shall be included in the protection scope of the present invention.

Claims (8)

1. A method of constructing a low-rank circulant matrix, characterized in that the construction method of the low-rank circulant matrix includes: utilizing isomorphism theory to reduce the search space of the circulant matrix; utilizing a rank algorithm to search to obtain circulant matrices of different ranks; The construction method of the low-rank circulant matrix includes: given the row number L and the row or column weight m of the circulant matrix C, constructing the circulant matrix C is equivalent to designing the non-zero element position set S={s ₁ of the first row, s ₂ ,s ₃ ,...,s _m }, that is, a set whose cardinality is m, construct a position set S={s ₁ ,s ₂ ,s ₃ ,...,s with the basis m _m }, where, for 1≤i＜j≤m, 0≤s _i ＜s _j≤L -1; It can be seen from the number and value range of elements in the set S that the total number of position sets S is: Any position set S is isomorphic to a position set S _- containing 0 elements: The subtraction operation in the set S _- is performed modulo L, and the element s ₁ in the position set S is directly set to 0. It can be seen from the non-repeatability of the position set that the total number of the position set S is reduced to: The elements (s ₂ -s ₁ ) in the set S _- are relatively prime with L. From the knowledge of number theory, it can be known that there is a number n such that (s ₂ -s ₁ )·n=1 (mod L), then, the set S _-Isomorphic to a position set S _* containing 0 elements and 1 elements, that is: The multiplication operation in the set S _* is performed modulo L. The element s ₁ in the position set S is directly set to 0 and the element s ₂ is set to 1. It can be seen from the non-repeatability of the position set that the total number of the position set S The number is reduced to: The minimum value of the circulant matrix rank is 1 and the maximum value is L. Set a threshold value R and only need to find a circulant matrix with a rank less than R. 2. The construction method of a low-rank circulant matrix as claimed in claim 1, characterized in that the construction method of the low-rank circulant matrix is such that _C1 and _C2 are two rows or columns with a weight of m and a size of L× The binary circulant matrix of L, their first row non-zero position sets are recorded as S ₁ ={s _1,1 ,s _1,2 ,s _1,3 ,...,s _1,m } and S ₂ respectively ={s _2,1 ,s _2,2 ,s _2,3 ,...,s _2,m }, if the circulant matrix C ₂ is obtained from C ₁ according to at least one of the following conditions, then C ₁ is said to be isomorphic to C ₂ , recorded as For constant c∈{0,1,2,...,L-1}, all elements of set S ₂ can be obtained by adding all elements of set S ₁ plus a constant c, that is, for 1≤i≤m, s _2,i =s _1,i +c(mod L), c∈{1,2,...,L-1}, and is relatively prime with L; All elements of set S ₂ and all elements of set S ₁ satisfy the following equation: for 1≤i≤m,s _2,i =c·s _1,i (mod L). 3. The construction method of a low-rank circulant matrix as claimed in claim 1, characterized in that the search method of the low-rank circulant matrix includes: Step 1: Select (m-1) elements from the set {1,2,...,L-1}, and select a set S of non-zero element positions in the order of combination; Step 2: Generate a circulant matrix C based on the position set S in step 1; Step 3: Calculate the rank r of the circulant matrix C in Step 2; Step 4: If r is less than R, store the location set S and record its rank as r; Step 5: Repeat steps 1 to 4 until all position sets with ranks from 1 to R are found, or all are found. set of locations. 4. A multivariate LDPC code construction method based on the construction method of a low-rank circulant matrix according to any one of claims 1 to 3, characterized in that the multivariate LDPC code construction method includes: a search method based on a low-rank circulant matrix and the algorithm for testing whether there is a 4-ring in the circulant matrix C, and obtain a binary circulant matrix C of size L×L, whose Tanner diagram has a girth of at least 6; replace the non-zero element 1 in the circulant matrix C with For the non-zero field elements on the finite field GF(q), during the replacement process, the obtained multivariate matrix is not of full rank. Directly replace the non-zero element 1 in the matrix C randomly with the non-zero field elements, then the obtained The multivariate matrices are basically all full-rank; using the non-zero domain element assignment method, based on a binary circulant matrix, a set of multivariate LDPC codes with flexible domain order and code rate are obtained. The rank of the binary circulant matrix C is R, the optional code rates of the multivariate LDPC code are 1-R, 1-R-1/L, 1-R-2/L, 1-R-3/L,...,0. 5. The multivariate LDPC code construction method as claimed in claim 4, characterized in that the algorithm for checking whether there is a 4-ring in the circulant matrix C by the multivariate LDPC code construction method includes: Step 1: Obtain the difference set D={ _{d∈Z L} |d=s i -s _j (mod L),1≤ based on the position set S={s ₁ , s ₂ , s ₃ ,..., s _m } i≤m,1≤j≤m,i≠j}; Step 2: Calculate the number of elements in the difference set D, or check whether the set E={e|e=d _i +d _j (mod L),i≠j,d _i ∈D,d _j ∈D,} There are zero elements; Step 3: If the number of elements in the difference set D is less than Or there are zero elements in the set E, and the direct output is that there is a 4-ring; otherwise, the output is that there is no 4-ring. 6. The multivariate LDPC code construction method as claimed in claim 4, characterized in that the assignment method of non-zero domain elements in the multivariate LDPC code construction method includes: Method 1: Replace all non-zero elements 1 in each column of the binary circulant matrix C with the same non-zero field element on the finite field GF(q). The non-zero field elements here are randomly selected to obtain a GF( The matrix C _q on q), the null space of the matrix C _q gives a set of q-element LDPC codes with code rate (1-R) and code length L; Method 2: Replace the non-zero element 1 of a certain column or columns in the binary circulant matrix C with a different non-zero field element on the finite field GF(q), and replace the non-zero element 1 of each remaining column with a finite The same non-zero field element on the field GF(q), the non-zero field elements are randomly selected, and a matrix C _q on the GF(q) is obtained. As the columns of the matrix C _q have columns with different non-zero field elements As the number gradually increases, the rank of matrix C _q will increase one by one until it is full rank. The null space of matrix C _q defines a set of q-element LDPC codes with variable code rates. 7. An image processing technology, characterized in that the image processing technology executes the low-rank circulant matrix construction method described in any one of claims 1 to 3. 8. A channel coding scheme, characterized in that the channel coding scheme implements the construction method of the multivariate LDPC code described in any one of claims 5 to 6.