CN111740747B - Construction method of low-rank cyclic matrix and associated multi-element LDPC code - Google Patents
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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
Technical Field
The application relates to the technical field of wireless communication technology 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 function stage of 5G standardization is finished, and the next stage of standardization mainly faces to the application scene of the main Internet of things/vertical industry, and a wireless communication transmission scheme supporting the information society of the next 10 years is provided. The standardization mainly comprises two aspects: high reliability low latency communication traffic (URLLC) and large-scale machine communication (mctc). Unlike 5G, the "everything-friendly" landscape of 6G needs to meet the requirements of real-time, reliability, throughput and mass connection at the same time, which will put forward a new challenge for the new generation of wireless communication networks. Delay and reliability indicators are often considered together to refer to the maximum transmission delay of a communication system at a certain probability of correct transmission. For channel coding, it is required that the coding processing delay is low and the error floor generated by the coding algorithm is eliminated. In combination with soft-output iterative decoding, LDPC codes are a competitive practical channel coding technique. Research shows that under the condition of medium and short code length, compared with binary LDPC codes under the condition of the same bit length, the multi-element LDPC codes have the following advantages: 1) There is more (1-1.3 dB) coding gain; 2) The burst error resistance is stronger; 3) More easily combined with higher order modulation. In recent years, under the framework of iterative decoding, 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, redundant rows of the LDPC code check matrix can accelerate the decoding convergence speed, thereby effectively reducing decoding delay. Furthermore, in image processing, the data matrix of natural images is typically low rank or near low rank. That is, each row (or column) of the matrices may be linearly represented by other rows (or columns), thereby containing a large amount of redundant information. Based on the redundant information, noise information of the image can be removed, and true image information can be recovered, and wrong image information can be recovered. However, there is relatively little research on low rank matrix construction. In summary, it is very interesting to study the construction method of low rank matrices (or matrices 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 number of cyclic matrices based on European geometry, reed-Solomon codes and two-dimensional maximum distance separable code structures is limited; in the computer poor search method based on cyclic codes and isomorphic theory, as the size and the weight of rows (or columns) of the matrix are increased, the search space is increased sharply, and searching and determining different isomorphic classes become difficult.
Through the above analysis, the problems and defects existing in the prior art are as follows: the number of the circulation matrixes constructed based on the algebraic structure is extremely limited, so that the actual application needs are difficult to meet; and 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, the poor search is extremely difficult to finish.
The difficulty of solving the problems and the defects is as follows: finding new algebraic structures to construct the circulant matrix is very difficult; when the matrix and row (column) weights are large, computer-based search space is large, poor search schemes are difficult to accomplish, and finding a different configuration of matrix is impractical.
The meaning of solving the problems and the defects is as follows: the application converts the search space of the matrix into the search space of the position set, and finds cyclic matrices with different ranks based on a rank calculation algorithm. This provides a wide variety of low rank circulants for image processing techniques. In addition, the application also provides a construction method of the multi-element LDPC code based on the assignment method of the multi-element domain elements. This provides a candidate coding scheme for high-reliability low-delay communication services in a new generation mobile communication system.
Disclosure of Invention
Aiming at the problems existing in the prior art, the application provides a construction method of a low-rank cyclic matrix and a multi-element LDPC code associated with the construction method.
The application is realized in such a way that a construction method of a low-rank cyclic matrix comprises the following steps: reducing the search space of the cyclic matrix by utilizing isomorphism theory; and searching by using a rank calculation algorithm to obtain cyclic matrixes with different ranks. A method of constructing a multi-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 C 1 And C 2 Binary cyclic matrices of size L x L, with two rows or columns of weight m, the first row non-zero sets of which are denoted S 1 ＝{s 1,1 ,s 1,2 ,s 1,3 ,...,s 1,m Sum S 2 ＝{s 2,1 ,s 2,2 ,s 2,3 ,...,s 2,m If the matrix C is cyclic 2 From C 1 Obtained according to at least one of the following conditions, then referred to as C 1 Isomorphic to C 2 Is marked as
For a constant c ε {0,1, 2., L-1}, set S 2 All elements of the set S 1 Is obtained by adding a constant c to all elements of (1. Ltoreq.i.ltoreq.m, s) 2,i ＝s 1,i +c (mod L), c ε {1,2,.., L-1}, and interl;
set S 2 All elements and sets S of (2) 1 All elements of (2) satisfy the following equation: for i is more than or equal to 1 and less than or equal to m, s 2,i ＝c·s 1,i (mod L)。
Further, the construction method of the low-rank cyclic matrix comprises the following steps: given the number L of rows and the row or column weight m of the circulant matrix C, constructing the circulant matrix C is equivalent to designing a non-zero element position set s= { S for the first row 1 ,s 2 ,s 3 ,...,s m A set of m bases, i.e. a set of positions s= { S with m bases is constructed 1 ,s 2 ,s 3 ,...,s m And (2) wherein, for 1.ltoreq.i < j.ltoreq.m, 0.ltoreq.s i ＜s j ≤L-1；
The number and the value range of the elements in the set S can be known, and the total number of the position set S is as follows:
any one position set S is isomorphic to one position set S containing 0 element - ：
Set S - The subtraction operation in (a) is performed under the mode L, and the element S in the position set S is directly processed 1 Let 0 be, the total number of the position sets S is reduced as follows from the unrepeatability of the position sets:
set S - Element(s) 2 -s 1 ) With L, as known from knowledge of number theory, there is a number n such that (s 2 -s 1 ) N=1 (mod L), then set S - Isomorphic to a set S of positions containing 0 element and 1 element * The method comprises the following steps:
set S * The multiplication operation in (a) is performed under the mode L, and the element S in the position set S is directly processed 1 Set to 0, element s 2 Let 1 be the non-repeatability of the set of positions, the total number of sets of positions S is reduced to:
the minimum value of the rank of the cyclic matrix is 1, the maximum value is L, and a threshold R is set, so that only the cyclic matrix with the rank smaller than R is needed to be searched.
Further, the search method of the low-rank cyclic matrix comprises the following steps:
step one, selecting (m-1) elements from the sets {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 ranks from 1 to R are found or all the position sets are foundA set of locations.
Another object of the present application is to provide a multi-LDPC code construction method based on the construction method of a low rank cyclic matrix, the multi-LDPC code construction method 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 4-rings exist in the cyclic matrix C, wherein the girth of a Tanner graph is at least 6; replacing the non-zero element 1 in the cyclic matrix C with a non-zero field element on the finite field GF (q), wherein in the replacing process, the obtained multi-element matrix is not full-rank, and the non-zero element 1 in the matrix C is directly replaced with the non-zero field element, so that the obtained multi-element 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 order and code rate are obtained based on a binary cyclic matrix, the rank of the binary cyclic matrix C is R, and the selectable code rate of the multi-element LDPC codes is 1-R,1-R-1/L,1-R-2/L, 1-R-3/L.
Further, the algorithm for checking whether 4-rings exist in the cyclic matrix C by the multi-LDPC code construction method includes:
step one, according to the position set s= { S 1 ,s 2 ,s 3 ,...,s m The difference set d= { D e Z } is obtained L |d＝s i -s j (mod L),1≤i≤m,1≤j≤m,i≠j}；
Step two, the number of elements in the difference set D is calculated, or the check set e= { e|e=d i +d j (modL),i≠j,d i ∈D,d j E, D, whether zero elements exist in the E;
step three, if the number of elements in the difference set D is smaller thanOr zero elements exist in the set E, and 4-rings exist in direct output; otherwise, the output does not have a 4-ring.
Further, the assignment method of the non-zero field elements of the multi-element LDPC code construction method comprises the following steps:
method 1: replacing all non-zero elements 1 of each column in the binary cyclic matrix C with the same one over the finite field GF (q)A non-zero field element, wherein the non-zero field element is randomly selected to obtain a matrix C over GF (q) q Matrix C q A group of q-element LDPC codes with code rate of (1-R) and code length of L are given out in the zero space of (2);
method 2: the non-zero element 1 of one or some columns in the binary cyclic matrix C is replaced by non-identical non-zero field elements on the finite field GF (q), and the non-zero element 1 of each remaining column is replaced by the same non-zero field element on the finite field GF (q), and the non-zero field elements are randomly selected to obtain the matrix C on the GF (q) q . Generally, with matrix C q The number of columns with different non-zero field elements in the columns gradually increases, matrix C q The ranks increase one by one until the rank is full, matrix C q A set of variable rate q-ary LDPC codes is defined.
Another object of the present application is to provide an image processing technology, including a matrix recovery and matrix decomposition technology, wherein the matrix recovery and matrix decomposition require a low rank matrix, and the construction method of the low rank cyclic matrix performs the following steps: reducing the search space of the cyclic matrix by utilizing isomorphism theory; and searching by using a rank calculation algorithm to obtain cyclic matrixes with different ranks.
Another object of the present application is to provide a channel coding scheme including a multi-LDPC code, the construction method of the multi-LDPC code performing the steps of: and obtaining the multi-element LDPC codes with different orders and different code rates by using a multi-element domain value giving method and the low-rank cyclic matrix construction method.
By combining all the technical schemes, the application 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 application obtains the low-rank cyclic matrix which is easy to realize by hardware. Isomorphism theory of the cyclic matrix is discussed, and a construction method of the low-rank cyclic matrix is provided based on the isomorphism theory. Considering the influence of a ring with the length of 4 in a Tanner graph on iterative decoding performance, a non-zero domain element assignment method is adoptedA multi-element LDPC code construction method under different orders is provided. Numerical simulation results show that compared with binary LDPC codes constructed based on PEG algorithm, the constructed multi-element LDPC codes have a codeword error rate of 10 in a 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 addition, the performance of the constructed multi-element LDPC code is almost consistent under the conditions of iteration 5 and 50 times, which provides an effective candidate coding scheme for low-delay and high-reliability communication.
The application adopts the channel coding technology of low-delay high-reliability communication, and the communication services are mainly oriented to the Internet of things represented by machine communication, and have the characteristics of small data packets, low power consumption, mass connection, strong burst property and the like, and a channel coding scheme with high coding and decoding speed, strong burst resistance and short code length is required. The application relates to a construction method of a low-rank cyclic matrix, wherein the cyclic matrix refers to a square matrix with a size, each row of the cyclic matrix is right (or left) cyclic shift of the previous row, and the first row is right (or left) cyclic shift of the last row; each column of it is a downward (or upward) cyclic shift of the column to its left, the first column being a downward (or upward) cyclic shift of the last column. Obviously, the row weight and column weight of the circulant matrix are the same.
According to the application, firstly, the isomorphism theory is utilized to reduce the search space of the cyclic matrix, and the cyclic matrices with different ranks are obtained by searching through a rank calculation algorithm. And directly searching the cyclic matrixes with different ranks by using a mode of calculating the ranks, and not searching and dividing isomorphic classes of the cyclic matrixes. Further, through a 4-ring (abbreviated as 4-ring) structure with the length of 4 in a cyclic matrix Tanner graph, 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 (Additive White Gaussian Noise, AWGN) channel, the constructed multi-element LDPC code has good iterative decoding performance, and performance curves at iteration 5 times and at iteration 50 times almost overlap.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings that are needed 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 other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
Fig. 1 is a flowchart of a method for constructing a low rank cyclic matrix according to an embodiment of the present application.
Fig. 2 is a schematic diagram showing performance comparison of error code rates of a (31, 15) LDPC code on GF (64) and a binary (186, 90) LDPC code constructed based on a PEG algorithm according to an embodiment of the present application.
Fig. 3 is a schematic diagram showing the performance comparison of the error rate 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 according to the embodiment of the present application.
Fig. 4 is a schematic diagram of codeword error rates of (31, 15) LDPC codes over GF (4), GF (8), GF (32), and GF (128) under 5 and 50 iterations according to an embodiment of the present application.
Detailed Description
The present application will be described in further detail with reference to the following examples in order to make the objects, technical solutions and advantages of the present application more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the application.
Aiming at the problems existing in the prior art, the application provides a construction method of a low-rank cyclic matrix and a multi-element LDPC code associated with the method, and the application is described in detail below with reference to the accompanying drawings.
As shown in fig. 1, the construction method of the low-rank cyclic matrix and the multi-element LDPC code provided by the present application includes the following steps:
s101: the search space of the cyclic matrix is reduced by utilizing isomorphism theory;
s102: searching by using a rank calculation algorithm to obtain cyclic matrixes with different ranks;
s103: obtaining a cyclic matrix without a 4-ring 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 cyclic matrix without 4-rings.
The application constructs a low-rank cyclic matrix and a multi-element LDPC code to give the following embodiment:
example 1, a cyclic matrix with a number of rows (or columns) of 31 and a row (or column) weight of 5 and a set of multi-element LDPC codes with a code length of 31, 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 are constructed.
Referring to fig. 1, the implementation steps of the present application 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 set as S= { S 1 ,s 2 ,s 3 ,s 4 ,s 5 And }, wherein 0.ltoreq.s for 1.ltoreq.i < j.ltoreq.5 i ＜s j And is less than or equal to 30. Obviously, the search space of the position set S isBased on isomorphism theory, the location set can be reduced to S * ＝{0,s 1 ,s 2 ,s 3 ,s 4 And }, wherein 1.ltoreq.s for 1.ltoreq.i < j.ltoreq.4 i ＜s j And is less than or equal to 30. Obviously, the set of positions S * Is +.>
Step 2, according to the position set S in step 1 * 3 cyclic matrices of different ranks are obtained by using a rank-solving algorithm, and the corresponding ranks are 16, the position sets are {0,1,3,7, 15}, the ranks are 21, the position sets are {0,1,2,6, 27}, the ranks are 26, and the position sets are {0,1,2,3,5}.
And 3, according to the 3 cyclic matrices obtained in the step 2, using a 4-ring calculation algorithm, obtaining two cyclic matrices without 4-rings, wherein the cyclic matrices are respectively provided with a rank of 16, a position set of {0,1,3,7, 15} and a rank of 26 and a position set of {0,1,2,3,5}.
Step 4, according to the cyclic matrix with the rank of 16 and the position set of {0,1,3,7, 15} obtained in the step 3, a q-ary LDPC code with the code length of 31 and the variable code rate can be obtained by using a domain element assignment method, wherein the selectable 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 and 1/31}.
The method for constructing a low rank cyclic matrix provided by the present application may be implemented by other steps by those skilled in the art, and the method for constructing a low rank cyclic matrix provided by the present application in fig. 1 is merely a specific embodiment.
The technical scheme of the application is further described below with reference to the accompanying drawings.
1. Low-rank cyclic matrix construction method based on isomorphism theory
1.1 circulant matrix and isomorphic theory thereof
Here, the cyclic matrix c= [ C ] to be considered by the present application i,j ] L×L Recorded as a binary matrix with a row (or column) weight of m and a size of L x L. Due to the cyclic shift characteristic of the cyclic matrix C, the method only needs to mark the positions of the non-zero elements of the first row of the cyclic matrix C. The non-zero element position set may be set to s= { S 1 ,s 2 ,s 3 ,...,s m And (2) wherein, for 1.ltoreq.i < j.ltoreq.m, 0.ltoreq.s i ＜s j And L-1 is not more than. Thus, 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. Nodes in the Tanner graph are divided into two classes: variable nodes (Variable nodes) (or code bit nodes) and Check nodes (Check nodes) (or constraint nodes) are denoted by VN and CN, respectively. The lines in the Tanner graph connect only nodes of different types. The Tanner graph of the circulant matrix C can be obtained as follows: element C in C i,j When the variable node is 1, the ith check node (CN i) is connected with the jth variable node (VNj); otherwise there is no wire connection between them. The length of the shortest loop in the Tanner graph of the circulant matrix C is called Girth (Girth). If the Tanner graph of the two circulants is isomorphic,the two circulants are also said to be isomorphic. According to the literature [ XU Hengzhou, BAI boom, ZHU Min, et al construction of short-block nonbinary LDPC codes based on cyclic codes [ J ]].China Communications，2017，14(8):1-9]The isomorphism theorem of the cyclic matrix is given below without evidence.
Theorem 1 (isomorphism theory of cyclic matrices): let C 1 And C 2 Binary cyclic matrix with weight of m and size of L×L for two rows (or columns), and the first row non-zero position set is respectively marked as S 1 ＝{s 1,1 ,s 1,2 ,s 1,3 ,...,s 1,m Sum S 2 ＝{s 2,1 ,s 2,2 ,s 2,3 ,...,s 2,m }. If the cyclic matrix C 2 Can be obtained from C 1 Obtained under at least one of the following conditions, then referred to as C 1 Isomorphic to C 2 Is marked as
For a constant c ε {0,1, 2., L-1}, set S 2 All elements of the set S 1 Is obtained by adding a constant c to all elements of (1. Ltoreq.i.ltoreq.m, s) 2,i ＝s 1,i +c (mod L). Let c be {1,2,., L-1}, and interpixel with L.
Set S 2 All elements and sets S of (2) 1 All elements of (2) satisfy the following equation: for i is more than or equal to 1 and less than or equal to m, s 2,i ＝c·s 1,i (mod L)。
1.2 Low rank cyclic matrix construction method based on isomorphism theory
Given the number L of rows and the row (or column) weight m of the circulant matrix C, constructing the circulant matrix C is equivalent to designing a non-zero element position set s= { S for the first row 1 ,s 2 ,s 3 ,...,s m I.e. a set of m for one base (carry). Therefore, the application mainly constructs a position set S= { S with a base of m 1 ,s 2 ,s 3 ,...,s m And (2) wherein, for 1.ltoreq.i < j.ltoreq.m, 0.ltoreq.s i ＜s j ≤L-1。
The number and the value range of the elements in the set S can be known, and the total number of the position set S is as follows:
from the first condition of theorem 1, it is known that any one of the position sets S is isomorphic to one position set S containing 0 elements - The method comprises the following steps:
note that set S - The subtraction operation in (2) is performed in modulo L. Thus, the element S in the position set S can be directly combined 1 Let 0 be, the total number of the position sets S is reduced as follows from the unrepeatability of the position sets:
this effectively reduces the search space of the set of locations S. Hypothesis set S - Element(s) 2 -s 1 ) With L, as known from knowledge of number theory, there is a number n such that (s 2 -s 1 ) N=1 (mod L). Then, as known from the second condition of theorem 1, set S - Isomorphic to a set S of positions containing 0 element and 1 element * The method comprises the following steps:
note that set S * The multiplication of (2) is performed in modulo L. In this case, the element S in the position set S can be directly used 1 Set to 0, element s 2 Let 1 be the non-repeatability of the set of positions, the total number of sets of positions S is reduced to:
this may further reduce the search space of the set of locations S. Due to practical requirements, the application only needs to construct a cyclic matrix with a specific rank. The minimum value of the cyclic matrix rank is 1 and the maximum value is L, as known from the size of the cyclic matrix. Since the application mainly focuses on low rank matrix, in order to reduce search space, a threshold R is set, and only a cyclic matrix with rank smaller than R is needed to be searched. A search algorithm for constructing a low rank cyclic matrix, algorithm 1, is given below.
Algorithm 1 cyclic matrix search algorithm with rank less than R
To demonstrate the effectiveness of algorithm 1, table 1 gives the search results for a portion of the low rank cyclic matrix.
Table 1 partial cyclic matrix search based on algorithm 1
2. Multi-element LDPC code construction 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 cyclic matrix is analyzed and a method of determining the 4-ring of the cyclic matrix is presented.
4-ring structure in cyclic matrix C:
the 4-ring in the cyclic matrix C consists of four 1 elements, which are distributed in two rows and two columns, and the structure of the 4-ring is shown in figure 1. From the cyclic matrix and the set of positions s= { S 1 ,s 2 ,s 3 ,...,s m The relationship 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 brackets of the above formula is based on modulo L. Clearly, the full requirement that this 4-ring exists in FIG. 2 is that one of the following formulas holds:
s i -s j ＝s k -s l (mod L) (1)
or alternatively
(s i -s j )+(s l -s k )＝0(mod L) (2)
Since i+.j, k+.l, (i, j) +.k, l, the two above formulas hold true in relation to two numbers, i.e.(s) i -s j ) Sum(s) l -s k ). Note that these two numbers are positive numbers obtained under the modulus L. Based on this, a new concept "difference set" is defined here in terms of the set of locations S.
Definition 1 (difference set of position set): let the position set of the cyclic matrix C be s= { S 1 ,s 2 ,s 3 ,...,s m And the number of rows (or columns) of the circulant matrix C is L. The difference set of the position set S is d= { D e 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 the set isBecause of the unrepeatability of the set, if the number of elements of the difference set D is less than +.>Then the difference is described as at leastThe two elements are equal, which also means that equation (1) is true. Alternatively, when the addition of two elements in the difference set D is equal to zero at the modulus L, then the equation (2) is true. This also corresponds to the presence of a 4-ring. This means that at least one 4-ring exists in the circulant matrix C. Based on this, the application herein presents an algorithm for checking the presence of 4-rings in the circulant matrix C, algorithm 2.
Algorithm 2 Algorithm to check for the presence of 4-rings in the circulant matrix C
According to algorithm 1 and algorithm 2, the application can obtain a low-rank cyclic matrix position set which does not contain a 4-ring. To demonstrate the effectiveness of algorithm 2, table 2 gives some sets of positions, which correspond to a cyclic matrix Tanner graph without a 4-ring.
Table 2 sets of positions of the circulant matrix excluding the 4-ring
2.2 construction method of multi-element LDPC code
The application mainly researches the construction method of the multi-element LDPC code. Based on algorithm 1 and algorithm 2, a binary cyclic matrix C of size lxl can be obtained, with a Tanner graph girth of at least 6. To construct the multi-element matrix, it is also necessary to replace the non-zero field element 1 in the cyclic matrix C with a non-zero field element on the finite field GF (q). Notably, during the replacement process, it is also ensured that the resulting multi-element matrix is not of full rank. Typically, non-zero element 1 in matrix C is randomly replaced by a non-zero field element directly, and the resulting multi-element matrix is substantially full rank. Therefore, the application adopts a non-zero domain element assignment method, and based on a binary cyclic matrix, a set of multi-element LDPC codes with flexible and variable domain order and code rate can be obtained. It is not necessary to assume that the rank of the binary cyclic matrix C is R, and then the optional code rate of the multi-element LDPC code proposed by the present application is 1-R,1-R-1/L,1-R-2/L, 1-R-3/L. The following briefly describes the assignment of two non-zero field elements.
Method 1: 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), where the non-zero field elements are randomly selected. Thus, a matrix C over GF (q) can be obtained q . From document [ XU Hengzhou, FENG Dan, SUN Cheng et al Algebraic-based nonbinary LDPC codes with flexible field orders and code rates [ J].China Communications，2017，14(4):111-119]The theorem 1 of (2) indicates that the binary cyclic matrix C and the multi-element matrix C q With the same rank. Thus, matrix C q A set of q-ary LDPC codes with code rate (1-R) and code length L is given in the zero space of (a).
Method 2: the non-zero elements 1 of a certain column (or columns) in the binary cyclic matrix C are replaced with non-zero field elements (different requirements) on the finite field GF (q), and the non-zero elements 1 of each of the remaining columns are replaced with the same non-zero field element on the finite field GF (q), where the non-zero field elements are randomly selected. Thus, a matrix C over GF (q) can be obtained q . Generally, with matrix C q The number of columns with different non-zero field elements in the columns gradually increases, matrix C q The ranks of (c) are increased one by one until the rank is full. Thus, matrix C q Can define a set of variable rate q-ary LDPC codes.
The technical effects of the present application will be described in detail with reference to simulation.
1. Simulation results
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 multi-element LDPC code is a multi-element sum-product algorithm (QSPA) based on Fast Fourier Transform (FFT). The selected higher order modulation is QPSK, 8PSK modulation and 64-QAM.
Consider a cyclic matrix with a number of rows (or columns) of 31 and a row (or column) weight of 5. From table 2, a cyclic matrix without 4-rings can be found, whose position matrix is {0,1,3,7, 15}, rank 16. According to method 1, it is possible toA group of q-element LDPC codes with a code length of 31 and a code rate of 15/31 are constructed. According to method 2, a set of variable rate q-ary LDPC codes with a code length of 31 can be obtained, the selectable rates of which 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, a 64-ary (31, 15) LDPC code can be obtained by selecting a finite field GF (64). Fig. 3 shows the codeword Error Rate (WER) performance of the code with QSPA iterated 1,3, 5 and 50 times. 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. Figure 3 also shows the codeword error rate performance and the finite length performance limit for the code at 186 bits code rate 15/31 with SPA iterated 5 and 50 times. 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-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 constructed 64-ary (31, 15) LDPC code has a small performance gap between 5 and 50 iterations; when the error rate is equal to 10 -5 The constructed 64-ary (31, 15) LDPC code is about 1dB from the finite length performance limit. Fig. 4 shows the codeword error rate performance of the constructed 64-bit (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 of the constructed multi-element code is greater than that of the two-element code, and the performance curves of the constructed multi-element code at iteration 5 and iteration 50 are almost overlapped. According to method 1, 4 (31, 15) multi-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 is at a codeword error rate of 10 -6 No error floor occurs. In addition, the proposed multi-element LDPC code can reach the decoding performance of 50 iterations only by 5 iterations.
The application adopts a construction method of a low-rank cyclic matrix. Firstly, the construction of a cyclic matrix is converted into the design of a non-zero element position set, and a search algorithm of a low-rank cyclic matrix is provided based on the isomorphism theory of the position set. Further, the 4-ring structure of the circulant matrix was analyzed, resulting in a circulant matrix having a girth of at least 6. Based on the above, two assignment methods of non-zero field elements are utilized, and a construction method of the multi-element LDPC code is provided. Numerical simulation results on the AWGN channel show that the constructed multi-element LDPC code has better decoding performance, and the decoding performance of iteration 50 times can be achieved only by iteration 5 times. This provides an efficient candidate coding scheme for low latency, high reliability wireless communications. To further enhance the performance of such codes, it is worthwhile to study how to optimize their non-zero field elements.
It should be noted that the embodiments of the present application can be realized in 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 special purpose design hardware. Those skilled in the art will appreciate that the above-described apparatus and methods may be implemented using computer-executable instructions and/or embodied in processor control code, such as provided on a carrier medium such as a magnetic disk, 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. The device of the present application and its modules may be implemented by hardware circuitry, 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., as well as software executed by various types of processors, or by a combination of the above hardware circuitry and software, such as firmware.
The foregoing is merely illustrative of specific embodiments of the present application, and the scope of the application is not limited thereto, but any modifications, equivalents, improvements and alternatives falling within the spirit and principles of the present application will be apparent to those skilled in the art within the scope of the present application.
Claims (8)
1. The construction method of the low-rank cyclic matrix is characterized by comprising the following steps of: reducing the search space of the cyclic matrix by utilizing isomorphism theory; searching by using a rank calculation algorithm to obtain cyclic matrixes with different ranks;
the construction method of the low-rank cyclic matrix comprises the following steps: given the number L of rows and the row or column weight m of the circulant matrix C, constructing the circulant matrix C is equivalent to designing a non-zero element position set s= { S for the first row 1 ,s 2 ,s 3 ,...,s m A set of m bases, i.e. a set of positions s= { S with m bases is constructed 1 ,s 2 ,s 3 ,...,s m And (2) wherein, for 1.ltoreq.i < j.ltoreq.m, 0.ltoreq.s i ＜s j ≤L-1；
The number and the value range of the elements in the set S can be known, and the total number of the position set S is as follows:
any one position set S is isomorphic to one position set S containing 0 element - ：
Set S - The subtraction operation in (a) is performed under the mode L, and the element S in the position set S is directly processed 1 Let 0 be, the total number of the position sets S is reduced as follows from the unrepeatability of the position sets:
set S - Element(s) 2 -s 1 ) With L mutual element, as known from knowledge of number theory, there is a number n, so thatGet(s) 2 -s 1 ) N=1 (mod L), then set S - Isomorphic to a set S of positions containing 0 element and 1 element * The method comprises the following steps:
set S * The multiplication operation in (a) is performed under the mode L, and the element S in the position set S is directly processed 1 Set to 0, element s 2 Let 1 be the non-repeatability of the set of positions, the total number of sets of positions S is reduced to:
the minimum value of the rank of the cyclic matrix is 1, the maximum value is L, and a threshold R is set, so that only the cyclic matrix with the rank smaller than R is needed to be searched.
2. The method for constructing a low rank cyclic matrix according to claim 1, wherein said method for constructing a low rank cyclic matrix is characterized in that C 1 And C 2 Binary cyclic matrices of size L x L, with two rows or columns of weight m, the first row non-zero sets of which are denoted S 1 ＝{s 1,1 ,s 1,2 ,s 1,3 ,...,s 1,m Sum S 2 ＝{s 2,1 ,s 2,2 ,s 2,3 ,...,s 2,m If the matrix C is cyclic 2 From C 1 Obtained under at least one of the following conditions, then referred to as C 1 Isomorphic to C 2 Is marked as
For a constant c ε {0,1, 2., L-1}, set S 2 All elements of the set S 1 Is obtained by adding a constant c to all elements of (1. Ltoreq.i.ltoreq.m, s) 2,i ＝s 1,i +c (mod L), c ε {1,2,.., L-1}, and interl;
set S 2 All elements and sets S of (2) 1 All elements of (2) satisfy the following equation: for i is more than or equal to 1 and less than or equal to m, s 2,i ＝c·s 1,i (mod L)。
3. The method for constructing a low rank cyclic matrix according to claim 1, wherein the method for searching the low rank cyclic matrix comprises:
step one, selecting (m-1) elements from the sets {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 ranks from 1 to R are found or all the position sets are foundA set of locations.
4. A multi-LDPC code construction method based on the construction method of a low rank cyclic matrix according to any one of claims 1 to 3, characterized in that the multi-LDPC code construction method comprises: 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 4-rings exist in the cyclic matrix C, wherein the girth of a Tanner graph is at least 6; replacing the non-zero element 1 in the cyclic matrix C with a non-zero field element on the finite field GF (q), wherein in the replacing process, the obtained multi-element matrix is not full-rank, and the non-zero element 1 in the matrix C is directly replaced with the non-zero field element, so that the obtained multi-element 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 order and code rate are obtained based on a binary cyclic matrix, the rank of the binary cyclic matrix C is R, and the selectable code rate of the multi-element LDPC codes is 1-R,1-R-1/L,1-R-2/L, 1-R-3/L.
5. The multi-LDPC code construction method of claim 4, wherein the algorithm for checking whether a 4-ring exists in the cyclic matrix C comprises:
step one, according to the position set s= { S 1 ,s 2 ,s 3 ,...,s m The difference set d= { D e Z } is obtained L |d＝s i -s j (mod L),1≤i≤m,1≤j≤m,i≠j}；
Step two, the number of elements in the difference set D is calculated, or the check set e= { e|e=d i +d j (mod L),i≠j,d i ∈D,d j E, D, whether zero elements exist in the E;
step three, if the number of elements in the difference set D is smaller thanOr zero elements exist in the set E, and 4-rings exist in direct output; otherwise, the output does not have a 4-ring.
6. The multi-LDPC code construction method of claim 4, wherein the multi-LDPC code construction method non-zero field element assignment method comprises:
method 1: replacing all non-zero elements 1 of each column in the binary cyclic matrix C 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 GF (q) q Matrix C q A group of q-element LDPC codes with code rate of (1-R) and code length of L are given out in the zero space of (2);
method 2: the non-zero element 1 of one or some columns in the binary cyclic matrix C is replaced by non-identical non-zero field elements on the finite field GF (q), and the non-zero element 1 of each of the remaining columns is replaced by the same non-zero field element on the finite field GF (q), wherein the non-zero field elements are as followsMechanically selecting to obtain a matrix C on GF (q) q With matrix C q The number of columns with different non-zero field elements in the columns gradually increases, matrix C q The ranks increase one by one until the rank is full, matrix C q A set of variable rate q-ary LDPC codes is defined.
7. An image processing technique, characterized in that the image processing technique runs the construction method of the low rank cyclic matrix according to any one of claims 1 to 3.
8. A channel coding scheme, characterized in that it runs the construction method of the multi-LDPC code as claimed in any one of claims 5 to 6.
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