CN111740747A

CN111740747A - Construction method of low-rank cyclic matrix and related multi-element LDPC code

Info

Publication number: CN111740747A
Application number: CN202010685467.XA
Authority: CN
Inventors: 徐恒舟; 朱海; 周慢杰; 车景平; 王娟娟
Original assignee: Zhoukou Normal University
Current assignee: Zhoukou Normal University
Priority date: 2020-07-16
Filing date: 2020-07-16
Publication date: 2020-10-02
Anticipated expiration: 2040-07-16
Also published as: CN111740747B

Abstract

The invention 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 an isomorphic theory; and searching by using a rank solving algorithm to obtain the cyclic matrixes with different ranks. The invention 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 construction method of the multi-element LDPC code under different orders is provided. Compared with the binary LDPC code constructed based on the PEG algorithm, the constructed multi-element LDPC code has the error code word rate of 10 under the BPSK modulation mode^‑5Nearby coding gain of 0.9 dB; when combined with high order modulation, there is a greater performance boost. In an additive white Gaussian noise channel, the constructed multivariate LDPC code has good iterative decoding performance, and performance curves at 5 times and 50 times of iteration almost overlap.

Description

Construction method of low-rank cyclic matrix and related multi-element LDPC code

Technical Field

The invention relates to the technical field of wireless communication and image processing, in particular to a construction method of a low-rank cyclic matrix and a related multi-element LDPC code thereof.

Background

At present, the basic functional stage of 5G standardization is completed, and the next stage of standardization mainly faces to the application scene of main internet of things/vertical industry, and provides a wireless communication transmission scheme supporting the information society of the next 10 years. The standardization mainly includes two aspects: high reliability low latency communication services (URLLC) and large scale machine communications (mtc). Unlike 5G, the 6G "everything is at will" vision needs to meet the requirements of real-time, reliability, throughput and massive connectivity at the same time, which will present a completely new challenge for new generation wireless communication networks. The delay and the reliability index are usually considered together, and refer to the maximum transmission delay of the communication system under a certain correct transmission probability. For channel coding, it is required to have a low coding/decoding processing delay and eliminate the error floor generated by the decoding algorithm. LDPC codes are a competitive practical channel coding technique in conjunction with soft-output iterative decoding. Research shows that, at medium and short code lengths, compared with a binary LDPC code at the same bit length, the multivariate LDPC code has the following advantages: 1) more (1-1.3 dB) coding gain is obtained; 2) the burst error resistance is stronger; 3) easier to combine with higher order modulation. In recent years, under an iterative decoding framework, the problem of high decoding complexity of the multi-element LDPC code is effectively solved, and a solid foundation is laid for the application of the multi-element LDPC code. In iterative decoding, the redundant row of the LDPC code check matrix can accelerate the decoding convergence speed, thereby effectively reducing the decoding time delay. Furthermore, in image processing, the data matrix of natural images is typically low rank or nearly low rank. That is, each row (or column) of these matrices may be linearly represented by other rows (or columns), thereby containing a large amount of redundant information. Based on the redundant information, the noise information of the image can be removed, the exact image information can be recovered, and the wrong image information can be recovered. However, relatively little research has been done on low rank matrix construction. In summary, it is very meaningful to study the construction method of the low rank matrix (or the matrix with more redundant rows).

The cyclic matrix has cyclic shift property, and is easy to realize in hardware based on a linear shift register. At present, the quantity of cyclic matrixes constructed based on Euclidean geometry, Reed-Solomon codes and two-dimensional maximum distance separable codes is limited; in the finite search method of the computer based on the cyclic code and isomorphic theory, as the size of the matrix and the weight of the row (or column) increase, the search space thereof increases sharply, and the search and the determination of different isomorphic classes become extremely difficult.

Through the above analysis, the problems and defects of the prior art are as follows: the number of the circulation matrixes constructed based on the algebraic structure is extremely limited, and the requirement of practical application is difficult to meet; the computer search algorithm based on the cyclic code and isomorphic theory has high operation complexity and large search space, and particularly when the matrix and the row (column) are heavy, it is extremely difficult to complete a poor search.

The difficulty in solving the above problems and defects is: finding a new algebraic structure to construct a circulant matrix is difficult; when the matrix and row (column) weight is large, the computer-based search space is large, it is difficult to implement a poor search scheme, and finding a different-structure matrix is not practical.

The significance of solving the problems and the defects is as follows: the invention converts the search space of the matrix into the search space of the position set, and finds the cyclic matrices with different ranks based on the rank solving algorithm. This provides a wide variety of low rank circulant matrices for image processing techniques. In addition, the invention also provides a construction method of the multi-element LDPC code based on the assignment method of the multi-element field elements. This provides a candidate coding scheme for high-reliability low-delay communication service in a new generation mobile communication system.

Disclosure of Invention

Aiming at the problems in the prior art, the invention provides a construction method of a low-rank cyclic matrix and a related multi-element LDPC code thereof.

The invention is realized in such a way that a construction method of a low-rank circulation matrix comprises the following steps: reducing the search space of the cyclic matrix by using isomorphic theory; and searching by using a rank solving algorithm to obtain the cyclic matrixes with different ranks. A method of constructing a multi-element LDPC code, the method of constructing an LDPC code comprising: and obtaining the multi-element LDPC codes with different orders and different code rates based on the multi-element domain assignment method and the low-rank cyclic matrix.

Further, the construction method of the low rank cyclic matrix is C₁And C₂Two binary circulant matrices with row or column weight m and size L × L, and their first row non-zero position sets are respectively denoted as S₁＝{s_1,1,s_1,2,s_1,3,...,s_1,mAnd S₂＝{s_2,1,s_2,2,s_2,3,...,s_2,mIf circulant matrix C₂From C₁Obtained by pressing at least one of the following conditions, then called C₁Is isomorphic in C₂Is marked as

For the constant c ∈ {0,1, 2., L-1}, the set S₂All elements of (2) can be formed from the set S₁Is added to a constant c, i.e., for 1. ltoreq. i.ltoreq.m, s_2,i＝s_1,i+ c (mod L), c ∈ {1,2,. and L-1}, and is reciprocal to L;

set S₂All elements of and set S₁Satisfies the following equation relationship: for 1. ltoreq. i.ltoreq.m, s_2,i＝c·s_1,i(mod L)。

Further, the method for constructing the low-rank cyclic matrix comprises the following steps: 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 S of the first row as { S }₁,s₂,s₃,...,s_mI.e. a set of m bases (Cardinality), constructing a set of m-based positions S ═ S₁,s₂,s₃,...,s_mJ is less than or equal to m and 0 is less than or equal to s for 1 and less than or equal to i_i＜s_j≤L-1；

The total number of the position set S is as follows according to the number and the value range of the elements in the set S:

any one ofEach position set S is isomorphic to a position set S containing 0 element_-：

Set S_-The subtraction operation in (1) is performed under a modulus L, and the element S in the position set S is directly collected₁Setting the position set as 0, and according to the irreproducibility of the position set, the total number of the position sets S is reduced to be:

set S_-Element(s) of (1)₂-s₁) With L, as known from the number theory, there is a number n such that(s)₂-s₁) N is 1(mod L), then the set S_-Isomorphism in a set S of positions containing 0 and 1 elements_*Namely:

set S_*The multiplication in (1) is performed under a modulus L, and the element S in the position set S is directly collected₁Set to 0, element s₂Setting the position set as 1, and according to the irreproducibility of the position set, the total number of the position sets S is reduced to be:

the minimum value of the rank of the cyclic matrix is 1, the maximum value is L, a threshold value R is set, and only the cyclic matrix with the rank smaller than R needs to be searched.

Further, the searching method of the low rank circulation matrix comprises the following steps:

step one, selecting (m-1) elements from a set {1, 2.,. L-1}, and selecting a group of non-zero element position sets S according to a combination sequence;

step two, generating a cyclic matrix C according to the position set S in the step one;

step three, calculating the rank r of the cyclic matrix C in the step two;

step four, if R is smaller than R, storing the position set S and recording the rank of the position set S as R;

step five, repeating the step one to the step four until all the position sets with the rank from 1 to R are found or all the position sets with the rank from 1 to R are found

A 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 comprising: obtaining a binary cyclic matrix C with the size of LxL based on a searching method of a low-rank cyclic matrix and an algorithm for checking whether a 4-ring exists in the cyclic matrix C, wherein the girth of a Tanner graph is at least 6; replacing non-zero elements 1 in the cyclic matrix C with non-zero field elements in a finite field GF (q), wherein in the replacement process, the obtained multivariate matrix is not full-rank, and the non-zero elements 1 in the matrix C are directly randomly replaced with the non-zero field elements, so that the obtained multivariate matrix is basically full-rank; a non-zero field element assignment method is adopted, a set of multi-element LDPC codes with flexibly variable field orders and code rates are obtained based on a binary cyclic matrix, the rank of the binary cyclic matrix C is R, and the selectable code rates of the multi-element LDPC codes are 1-R,1-R-1/L,1-R-2/L, 1-R-3/L.

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

step one, according to the position set S ═ S₁,s₂,s₃,...,s_mGet difference D ═ D ∈ Z_L|d＝s_i-s_j(modL),1≤i≤m,1≤j≤m,i≠j}；

Step two, calculating the number of elements in the difference set D, or checking the set E ═ E ═ D_i+d_j(modL),i≠j,d_i∈D,d_j∈ D, whether there is a zero element in the } step;

step three, if the number of elements in the difference set D is more than oneNumber less than

Or zero elements exist in the set E, and 4-rings exist in direct output; otherwise, the output has no 4-loops.

Further, the assignment method of the non-zero field element of the multivariate LDPC code construction method comprises the following steps:

the method comprises the following steps: all non-zero elements 1 of each column in the binary cyclic matrix C are replaced by the same non-zero field element on the finite field GF (q), wherein the non-zero field element is randomly selected to obtain a matrix C on the GF (q)_qThe matrix C_qThe null space of (A) gives a group of q-element LDPC codes with code rate of (1-R) and code length of L;

the method 2 comprises the following steps: replacing non-zero elements 1 of a certain column or some columns in the binary cyclic matrix C with different non-zero field elements on the finite field GF (q), replacing the non-zero elements 1 of each remaining column with the same non-zero field element on the finite field GF (q), wherein the non-zero field elements are randomly selected to obtain a matrix C on the GF (q)_q. In general, with matrix C_qThe number of columns with different non-zero field elements in the column increases gradually, matrix C_qWill increase one by one until full rank, matrix C_qDefines a set of variable rate q-ary LDPC codes.

It is another object of the present invention to provide an image processing technique, which includes matrix recovery and matrix decomposition techniques, wherein the matrix recovery and matrix decomposition require a low-rank matrix, and the method for constructing the low-rank cyclic matrix performs the following steps: reducing the search space of the cyclic matrix by using isomorphic theory; and searching by using a rank solving algorithm to obtain the cyclic matrixes with different ranks.

Another object of the present invention is to provide a channel coding scheme, the channel coding scheme including a multi-element LDPC code, the construction method of the multi-element LDPC code performing the following steps: and obtaining the multi-element LDPC codes with different orders and different code rates by using a multi-element domain value assigning method and the low-rank cyclic matrix constructing method.

All the technical schemes combined with the aboveThe invention has the advantages and positive effects that: in image processing, redundant information of a low-rank matrix can be used for image recovery and image feature extraction, and in iterative decoding, redundant rows of a check matrix can accelerate decoding convergence speed. The invention obtains the low-rank cyclic matrix which is easy to realize by hardware. Isomorphic theory of the cyclic matrix is discussed, and a construction method of the low-rank cyclic matrix is provided based on the isomorphic theory. Considering the influence of the ring with the length of 4 in the Tanner graph on the iterative decoding performance, a non-zero field element assignment method is adopted, and a construction method of the multi-element LDPC code under different orders is provided. The numerical simulation result shows that compared with the binary LDPC code constructed based on the PEG algorithm, the constructed multi-element LDPC code has the code error rate of 10 under the BPSK modulation mode^-5Nearby coding gain of 0.9 dB; when combined with high order modulation, there is a greater performance boost. In addition, the performance of the constructed multi-element LDPC code is almost consistent under the iteration of 5 times and 50 times, which provides an effective candidate coding scheme for low-delay and high-reliability communication.

The invention adopts a channel coding technology of low time delay and high reliability communication, the communication services mainly face to the Internet of things represented by machine communication, and the invention has the characteristics of small data packet, low power consumption, massive connection, strong burstiness and the like, and needs a channel coding scheme with high coding and decoding speed, strong burst resistance and short code length. In the method for constructing the low-rank cyclic matrix, the cyclic matrix refers to a square matrix with the size of one, each row of the square matrix is the right (or left) cyclic shift of the previous row, and the first row is the right (or left) cyclic shift of the last row; each column of it is a cyclic shift down (or up) of the column to its left, the first column is a cyclic shift down (or up) of the last column. Obviously, the row weight and column weight of the circulant matrix are the same.

The invention firstly utilizes the isomorphic theory to reduce the search space of the cyclic matrix and utilizes the rank-solving algorithm to search and obtain the cyclic matrices with different ranks. The method for calculating the rank is used for directly searching the cyclic matrix with different ranks, and isomorphic classes of the cyclic matrix are not searched and divided. Further, a ring (abbreviated as 4-ring) structure with the length of 4 in a circular matrix Tanner graph is adopted, an algorithm for determining the 4-ring is provided, a non-zero-element assignment method is also provided, and a multi-element LDPC code construction method with the girth of at least 6 is provided. Numerical simulation results show that in an Additive White Gaussian Noise (AWGN) channel, the constructed multivariate LDPC code has good iterative decoding performance, and performance curves at 5 iterations and 50 iterations almost overlap.

Drawings

In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings used in the embodiments of the present application will be briefly described below, and it is obvious that the drawings described below are only some embodiments of the present application, and it is obvious for those skilled in the art that other drawings can be obtained according to the drawings without creative efforts.

Fig. 1 is a flowchart of a method for constructing a low rank circulant matrix according to an embodiment of the present invention.

Fig. 2 is a schematic diagram illustrating comparison of code error rate performances of a (31, 15) LDPC code over GF (64) and a binary (186, 90) LDPC code constructed based on a PEG algorithm under different iteration numbers according to an embodiment of the present invention.

Fig. 3 is a schematic diagram illustrating a comparison of the error code rate performance of a (31, 15) LDPC code over GF (64) and a binary (186, 90) LDPC code constructed based on the PEG algorithm under high-order modulation according to an embodiment of the present invention.

Fig. 4 is a schematic diagram of the code error rate performance of the (31, 15) LDPC code over GF (4), GF (8), GF (32), and GF (128) at iterations 5 and 50 times provided in the embodiment of the present invention.

Detailed Description

In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

In view of the problems in the prior art, the present invention provides a method for constructing a low rank circulant matrix and a related multi-element LDPC code thereof, and the present invention is described in detail below with reference to the accompanying drawings.

As shown in fig. 1, the method for constructing a low-rank cyclic matrix and a multivariate LDPC code provided by the present invention comprises the following steps:

s101: the search space of the cyclic matrix is reduced by utilizing isomorphic theory;

s102: searching by using a rank solving algorithm to obtain cyclic matrices of different ranks;

s103: obtaining a 4-ring-free cyclic matrix by using a 4-ring calculation algorithm;

s104: and obtaining the multi-element LDPC codes with different orders and different code rates by using a domain element assignment method and a 4-ring-free cyclic matrix.

The invention constructs a low-rank cyclic matrix and a multi-element LDPC code to give the following embodiment:

embodiment 1, a circulant matrix with 31 rows (or columns) and 5 rows (or columns) is constructed, and a set of multi-element LDPC codes with 31 code length, 15/31 code rate, 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.

Referring to fig. 1, the implementation steps of the invention are as follows:

step 1, according to the cyclic matrix with 31 rows (or columns) and 5 rows (or columns) to be constructed, the corresponding position set is not set to be S ═ S₁,s₂,s₃,s₄,s₅J < 5 for 1. ltoreq. i, 0. ltoreq. s_i＜s_jLess than or equal to 30. Obviously, the search space for the set of locations S is

Based on isomorphic theory, the position set can be simplified into S^*＝{0,s₁,s₂,s₃,s₄J is less than 4 for 1 < i, and s is less than 1_i＜s_jLess than or equal to 30. Obviously, set of positions S^*The search space of

Step 2, according to the position set S in the step 1^*The search space of (2) is obtained by using a rank-solving algorithm to obtain 3The cyclic matrices of different ranks have a rank of 16 and a position set of {0,1, 3, 7, 15}, a rank of 21 and a position set of {0,1,2, 6, 27}, and a rank of 26 and a position set of {0,1,2, 3, 5 }.

And 3, obtaining two cyclic matrices without 4-loops by using a 4-loop calculation algorithm according to the 3 cyclic matrices obtained in the step 2, wherein the two cyclic matrices have the corresponding rank of 16, the position set of {0,1, 3, 7 and 15} and the rank of 26, and the position set of {0,1,2, 3 and 5 }.

And 4, obtaining a group of variable-code-rate q-element LDPC codes with the code length of 31 by utilizing a domain element assignment method according to the cyclic matrix with the rank of 16 and the position set of {0,1, 3, 7 and 15} obtained in the step 3, wherein the selectable code rates of the variable-code-rate q-element LDPC codes 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 and 1/31 }.

The method for constructing a low-rank cyclic matrix provided by the present invention can be implemented by using other steps by those skilled in the art, and the method for constructing a low-rank cyclic matrix provided by the present invention in fig. 1 is only a specific embodiment.

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

1. Isomorphic theory-based low-rank circulation matrix construction method

1.1 circulant matrix and isomorphic theory thereof

Here, the circulant matrix C ═ C considered in the present invention_i,j]_L×LThe invention can mark the first row of the circulant matrix C with non-zero element position only by marking the position of the first row of the circulant matrix C due to the cyclic shift property of the circulant matrix C₁,s₂,s₃,...,s_mJ is less than or equal to m and 0 is less than or equal to s for 1 and less than or equal to i_i＜s_jLess than or equal to L-1. The construction of the circulant matrix C is therefore equivalent to the design of the first row of the set S of non-zero element positions.

The Tanner Graph of circulant C is a Bipartite Graph (Bipartite Graph). The nodes in the Tanner graph are divided into twoClass (c): variable nodes (or coded bit nodes) and Check nodes (Check nodes) (or constraint nodes), denoted by VN and CN, respectively. The lines in the Tanner graph only connect nodes of different types. The Tanner graph of the circulant matrix C can be obtained by: when element C in C_i,jWhen the number is 1, the ith check node (CN i) is connected with the jth variable node (VNj); otherwise there is no wired connection between them. The length of the shortest loop in the Tanner graph of circulant matrix C is called Girth (Girth). Two circulants are said to be isomorphic if their Tanner graphs are isomorphic. According to the literature [ XU Hengzhou, BAI coding, ZHU Min, et al].China Communications，2017，14(8):1-9]Theorem 2 in (1), the isomorphic theorem for the circulant matrix is given below without proof.

Theorem 1 (isomorphic theory of circulant matrix): let C₁And C₂Two binary circulant matrixes with m in rows (or columns) and L × L in size, and the first row non-zero position sets of the binary circulant matrixes are respectively marked as S₁＝{s_1,1,s_1,2,s_1,3,...,s_1,mAnd S₂＝{s_2,1,s_2,2,s_2,3,...,s_2,m}. If the circulant matrix C₂Can be composed of C₁Obtained under at least one of the following conditions, then is called C₁Is isomorphic in C₂Is marked as

For the constant c ∈ {0,1, 2., L-1}, the set S₂All elements of (2) can be formed from the set S₁Is added to a constant c, i.e., for 1. ltoreq. i.ltoreq.m, s_2,i＝s_1,i+ c (mod L), assuming c ∈ {1, 2., L-1}, and being reciprocal to L.

1.2 isomorphic theory-based low-rank circulation matrix construction method

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 S of the first row { S }₁,s₂,s₃,...,s_mI.e. a set of bases (Cardinality) m. Therefore, the present invention mainly constructs a position set S ═ S based on m₁,s₂,s₃,...,s_mJ is less than or equal to m and 0 is less than or equal to s for 1 and less than or equal to i_i＜s_j≤L-1。

as can be seen from the first condition of theorem 1, any position set S is isomorphic to a position set S containing 0-element_-Namely:

note that set S_-The subtraction operation in (1) is performed modulo L. Thus, the elements S in the position set S can be directly grouped₁Assuming that 0 is set, the total number of position sets S is reduced by:

this effectively reduces the search space for the location set S. Set of assumptions S_-Element(s) of (1)₂-s₁) With L, as known from the number theory, there is a number n such that(s)₂-s₁) N is 1(mod L). Then, as can be seen from the second condition of theorem 1, the set S_-Isomorphism in a set S of positions containing 0 and 1 elements_*Namely:

note that set S_*The multiplication in (1) is performed modulo L. In this case, the element S in the position set S can be directly set₁Set to 0, element s₂Assuming 1, the total number of location sets S is reduced to:

this may further reduce the search space for the location set S. Due to practical requirements, the invention only needs to construct a cyclic matrix with a specific rank. From the size of the circulant matrix, the minimum value of the circulant matrix rank is 1, and the maximum value is L. Since the invention mainly focuses on low-rank matrix, in order to reduce search space, a threshold value R is set here, and only a cyclic matrix with rank less than R needs to be searched. A search algorithm for constructing a low rank cyclic matrix, algorithm 1, is given below.

Algorithm 1 rank less than R cyclic matrix search algorithm

To demonstrate the effectiveness of algorithm 1, table 1 presents search results for a portion of the low rank circulant matrix.

TABLE 1 partial circulant matrix based on Algorithm 1 search

2. Multi-element LDPC code structure based on low-rank cyclic matrix

2.1 4-Ring Structure of Cyclic matrix

Short loops, especially 4-loops, can degrade the iterative decoding performance of LDPC codes. Thus, the 4-ring structure of the circulant matrix is analyzed and a method of determining the 4-ring of the circulant matrix is presented.

4-ring structure in circulant matrix C:

the 4-ring in the circulant matrix C is composed of four 1 elements, which are distributed in two rows and two columns, and the structure is shown in fig. 1. From the set of circulant matrices and positions S ═ S₁,s₂,s₃,...,s_mThe relation between these four 1-element row and column coordinates can be abbreviated as:

(a_,s_i+a),(b,s_k+b),(b,s_l+b),(a,s_j+a)；

wherein a is more than or equal to 0 and less than or equal to b and less than or equal to L-1, i is more than or equal to 1 and less than or equal to j and less than or equal to m, and k is more than or equal to 1 and less than or equal to L and less than or equal to m. Note that the addition operation in parentheses in the above equation is performed based on modulo L. It is clear that a sufficient requirement for the presence of this 4-ring in fig. 2 is that one of the following equations 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 equations hold or not is related to two numbers, i.e.(s)_i-s_j) And(s)_l-s_k). Note that these two numbers are positive numbers obtained under modulo L. Based on this, a new concept "difference set" is defined here based on the set of locations S.

Define 1 (difference set of position set): let the position set of the circulant matrix C be S ═ S₁,s₂,s₃,...,s_mD, and the number of rows (or columns) of the cyclic 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. In theory, the basis of the set is

If the number of elements in the difference set D is less than or equal to the number of elements in the difference set D due to the non-repeatability of the set

It is stated that at least two elements in the difference set are equal, which also means that equation (1) holds. Alternatively, equation (2) holds when the sum of two elements in difference set D equals zero modulo L. This also corresponds to the presence of a 4-ring. This indicates that at least one 4-loop exists in the circulant matrix C. Based on this, the present invention provides an algorithm for checking whether a 4-loop exists in the circulant matrix C, i.e., algorithm 2.

Algorithm 2 Algorithm to check if there are 4-loops in circulant matrix C

According to algorithm 1 and algorithm 2, the invention can obtain a plurality of low rank cyclic matrix position sets without 4-rings. To demonstrate the effectiveness of algorithm 2, table 2 gives some sets of positions that correspond to no 4-loops in the circulant Tanner graph.

TABLE 2 set of circulant matrix positions that do not contain a 4-ring

2.2 construction method of multi-element LDPC code

The invention mainly researches a construction method of a multi-element LDPC code. Based on algorithm 1 and algorithm 2, a binary circulant matrix C of size lxl can be obtained, whose Tanner graph has a girth of at least 6. In order to construct the multivariate matrix, it is also necessary to replace non-zero element 1 in the circulant matrix C with a non-zero field element on the finite field gf (q). It is noted that in the replacement process, it is also ensured that the resulting multivariate matrix is not full rank. Typically, non-zero elements 1 in matrix C are directly randomly replaced by non-zero field elements, and the resulting multivariate matrix is then substantially all full rank. Therefore, the invention adopts a non-zero field element assignment method and can obtain a set of multi-element LDPC codes with flexible and variable field order and code rate based on a binary cyclic matrix. The binary cyclic matrix C is assumed to have a rank R, and the selectable code rates of the multi-element LDPC code are 1-R,1-R-1/L,1-R-2/L, 1-R-3/L. Two methods of assigning non-zero field elements are briefly described below.

The method comprises the following steps: all non-zero elements 1 of each column in the binary circulant matrix C are replaced by the same non-zero field element in the finite field gf (q), where the non-zero field elements are randomly selected. Thus, a matrix C over GF (q) is obtained_q. From the literature [ XU Hengzhou, FENG Dan, SUN Cheng, et al].China Communications，2017，14(4):111-119]From theorem 1, it can be seen that the binary cyclic matrix C and the multivariate matrix C_qWith the same rank. Thus, matrix C_qThe null space of (a) gives a set of q-element LDPC codes with code rate of (1-R) and code length of L.

The method 2 comprises the following steps: the non-zero elements 1 of a certain column (or some columns) in the binary cyclic matrix C are replaced by different non-zero field elements (different requirements) in the finite field gf (q), while the non-zero elements 1 of each remaining column are replaced by the same non-zero field element in the finite field gf (q), where the non-zero field elements are randomly selected. Thus, a matrix C over GF (q) is obtained_q. In general, with matrix C_qThe number of columns with different non-zero field elements in the column increases gradually, matrix C_qThe rank of (c) is increased one by one until the full rank. Thus, matrix C_qMay define a set of variable rate q-ary LDPC codes.

The technical effects of the present invention will be described in detail with reference to simulations.

1. Simulation result

The following simulation parameters are AWGN channel and BPSK modulation. The decoding algorithm of the binary LDPC code is a sum-product algorithm (SPA), and the decoding algorithm of the multivariate LDPC code is a Fast Fourier Transform (FFT) -based multivariate sum-product algorithm (QSPA). The selected high-order modulation is QPSK, 8PSK modulation and 64-QAM.

Consider a circulant matrix with 31 rows (or columns) and a row (or column) weight of 5. From table 2, a circulant matrix without 4-loops can be found, with a position matrix of 0,1, 3, 7, 15 and a rank of 16. According to method 1, a set of q-ary LDPC codes with a code length of 31 and a code rate of 15/31 may be constructed. According to the method 2, a group of q-element LDPC codes with the code length of 31 and variable code rates can be obtained, and the selectable 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 and 1/31 }. According to method 1, a 64-ary (31, 15) LDPC code can be obtained by selecting the finite field GF (64). Fig. 3 shows the Word Error Rate (WER) performance of the code under QSPA with 1, 3, 5 and 50 iterations. For comparison under the same code parameters (equivalent bit code length and code rate), a binary (186, 90) LDPC code is constructed here based on the PEG algorithm. Fig. 3 also shows the code error rate performance of the code for SPAs employing 5 and 50 iterations and the limited long performance limit of 186 bits for code length and 15/31 for code rate. It can be seen that when the number of iterations is 50 and the error word rate is equal to 10^-5The constructed 64-ary (31, 15) LDPC code has about 0.9dB coding gain over the binary (186, 90) LDPC code, and the performance gap is about 1.8dB when the number of iterations is 5. In addition, it can be seen that the performance gap between 5 and 50 iterations of the constructed 64-element (31, 15) LDPC code is small; when the code error rate is equal to 10^-5The constructed 64-element (31, 15) LDPC code is about 1dB from the finite length performance limit. Fig. 4 shows the error code rate performance of the constructed 64-element (31, 15) LDPC code and the binary (186, 90) LDPC code under high-order modulation. It can be seen that as the modulation order increases, the performance gap of the constructed multi-element code is larger than that of the binary code, and the performance curves of the constructed multi-element code are almost overlapped after 5 times and 50 times of iteration. According to method 1, 4 (31, 15) multi-element LDPC codes can be obtained by selecting finite fields GF (4), GF (8), GF (32) and GF (128). Fig. 4 shows the codeword error rate performance of these 4 codes at QSPA iterated 5 and 50 times. As can be seen from FIG. 4, the constructed multi-element LDPC code has better decoding performance and the error code rate is 10^-6There is no false floor. This is achieved byIn addition, the proposed multivariate LDPC code only needs 5 iterations to achieve decoding performance of 50 iterations.

The invention adopts a construction method of a low-rank circulation matrix. Firstly, the construction of the cyclic matrix is converted into the design of a non-zero element position set, and a search algorithm of the low-rank cyclic matrix is provided based on the isomorphic theory of the position set. Further, the 4-loop structure of the circulant matrix is analyzed to obtain the circulant matrix with the girth of at least 6. Based on the method, a construction method of the multi-element LDPC code is provided by utilizing two assignment methods of non-zero field elements. Numerical simulation results on the AWGN channel show that the constructed multi-element LDPC code has better decoding performance, and the decoding performance of 50 times of iteration can be achieved only by 5 times of iteration. This provides an efficient candidate coding scheme for low latency, high reliability wireless communications. To further improve the performance of such codes, it is worthwhile to optimize their non-zero field elements.

It should be noted that the embodiments of the present invention can be realized by hardware, software, or a combination of software and hardware. The hardware portion may be implemented using dedicated logic; the software portions may be stored in a memory and executed by a suitable instruction execution system, such as a microprocessor or specially designed hardware. Those skilled in the art will appreciate that the apparatus and methods described above may be implemented using computer executable instructions and/or embodied in processor control code, such code provided on a carrier medium such as a diskette, CD-or DVD-ROM, a programmable memory such as read-only memory (firmware), or a data carrier such as an optical or electronic signal carrier, for example. The apparatus 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, or programmable hardware devices such as field programmable gate arrays, programmable logic devices, etc., or by software executed by various types of processors, or by a combination of hardware circuits and software, e.g., firmware.

The above description is only for the specific embodiments of the present invention, but the scope of the present invention is not limited thereto, and any modification, equivalent replacement, and improvement made by those skilled in the art within the technical scope of the present invention disclosed in the present invention should be covered within the scope of the present invention.

Claims

1. A method for constructing a low-rank cyclic matrix is characterized by comprising the following steps: reducing the search space of the cyclic matrix by using isomorphic theory; and searching by using a rank solving algorithm to obtain the cyclic matrixes with different ranks.

2. The method of claim 1, wherein the low-rank cyclic matrix is constructed by a method of C₁And C₂Two binary circulant matrices with row or column weight m and size L × L, and their first row non-zero position sets are respectively denoted as S₁＝{s_1,1,s_1,2,s_1,3,...,s_1,mAnd S₂＝{s_2,1,s_2,2,s_2,3,...,s_2,mIf circulant matrix C₂From C₁Obtained by at least one of the following conditions, then is called C₁Is isomorphic in C₂Is marked as

For the constant c ∈ {0,1, 2., L-1}, the set S₂All elements of (2) can be formed from the set S₁Is added to a constant c, i.e., for 1. ltoreq. i.ltoreq.m, s_2,i＝s_1,i+ c (modL), c ∈ {1, 2.., L-1}, and is interdependent with L;

set S₂All elements of and set S₁Satisfies the following equation relationship: for 1. ltoreq. i.ltoreq.m, s_2,i＝c·s_1,i(modL)。

3. The method of constructing a low rank circulant matrix of claim 1, wherein the method of constructing a low rank circulant matrix comprises: to giveDetermining the row number L and the row or column weight m of the circulant matrix C, and constructing the circulant matrix C is equivalent to designing a non-zero element position set S ═ S of the first row₁,s₂,s₃,...,s_mI.e. a set of m bases (Cardinality), constructing a set of m-based positions S ═ S₁,s₂,s₃,...,s_mJ is less than or equal to m and 0 is less than or equal to s for 1 and less than or equal to i_i＜s_j≤L-1；

any position set S is isomorphic to a position set S containing 0 element_-：

set S_-Element(s) of (1)₂-s₁) With L, as known from the number theory, there is a number n such that(s)₂-s₁) N is 1(modL), then the set S_-Isomorphism in a set S of positions containing 0 and 1 elements_*Namely:

4. The method of constructing a low rank circulant matrix of claim 1, wherein the method of searching for a low rank circulant matrix comprises:

step three, calculating the rank r of the cyclic matrix C in the step two;

A set of locations.

5. A construction method of a multi-element LDPC code based on the construction method of the low rank cyclic matrix of any one of claims 1 to 4, the construction method of the multi-element LDPC code comprising: obtaining a binary cyclic matrix C with the size of L multiplied by L based on a searching method of a low-rank cyclic matrix and an algorithm for checking whether a 4-ring exists in the cyclic matrix C, wherein the girth of a Tanner graph is at least 6; replacing non-zero elements 1 in the cyclic matrix C with non-zero field elements on a finite field GF (q), wherein in the replacement process, the obtained multivariate matrix is not full-rank, and the non-zero elements 1 in the matrix C are directly randomly replaced with the non-zero field elements, so that the obtained multivariate matrix is basically full-rank; a non-zero field element assignment method is adopted, a set of multi-element LDPC codes with flexibly variable field orders and code rates are obtained based on a binary cyclic matrix, the rank of the binary cyclic matrix C is R, and the selectable code rates of the multi-element LDPC codes are 1-R,1-R-1/L,1-R-2/L, 1-R-3/L.

6. The multi-element LDPC code construction method of claim 5 wherein the algorithm for the multi-element LDPC code construction method to check whether a 4-loop exists in a circulant matrix C comprises:

step three, if the number of elements in the difference set D is less than

7. The method of constructing a multi-ary LDPC code according to claim 5 wherein the method of assigning non-zero field elements of the multi-ary LDPC code construction method comprises:

the method 2 comprises the following steps: the non-zero elements 1 of one or some columns in the binary circulant matrix C are replaced by non-identical non-zero field elements over the finite field gf (q), while the non-zero elements of each of the remaining columns1 is replaced by the same non-zero field element on the finite field GF (q), the non-zero field element is randomly selected to obtain a matrix C on the finite field GF (q)_qFollowing the matrix C_qThe number of columns with different non-zero field elements in the column increases gradually, matrix C_qWill increase one by one until full rank, matrix C_qDefines a set of variable rate q-ary LDPC codes.

8. An image processing technique, wherein the image processing technique operates the method of constructing a low-rank circulant matrix as claimed in any one of claims 1 to 4.

9. A channel coding scheme, characterized in that the channel coding scheme operates the construction method of the multi-element LDPC code of any one of claims 6 to 7.