Eight-ring QC-LDPC code construction method based on generalized arithmetic progression

Abstract

The invention relates to an eight-ring QC-LDPC (Quasi-Cyclic Low-Density Parity-Check) code construction method based on a generalized arithmetic progression. According to the construction method, first two rows of sequences and a first element of a third row are fixed through algebraic calculation of a generalized arithmetic progression, then other elements of the third row are obtained through a search algorithm based on the generalized arithmetic progression, and an exponential matrix meeting the condition without 4 and 6 rings is formed; and finally, the index matrix is correspondingly expanded to obtain a parity check matrix of the index matrix, so that four rings and six rings do not exist in a Tanner graph of the check matrix, the required storage space is small, and hardware implementation is easy. A simulation result shows that compared with an SS-QC-LDPC code, a DDS-QC-LDPC code and an SD-QC-LDPC code which are the same in code rate and code length, the net coding gain of the GASS-QC-LDPC code constructed by the method is improved to a certain degree, and the constructed GASS-QC-LDPC code has better waterfall region performance in a high signal-to-noise ratio region and does not have an obvious error leveling phenomenon.

Description

Technical Field

[0001] The present invention belongs to the technical field of channel coding, and relates to a method for constructing an 8-ring quasi-cyclic low-density parity-check (QC-LDPC) code based on a generalized arithmetic progression. Background Art

[0002] The basic purpose of communication is to effectively and reliably transmit the information generated by the source to the destination. However, in the process of information transmission, it is inevitable to be subject to various interferences, which will lead to the occurrence of bit errors in communication, reducing the effectiveness and reliability of information transmission. Therefore, channel coding technology is needed to ensure the reliable transmission of information.

[0003] QC-LDPC code is a typical linear block error correction code. Its parity check matrix (PCM) has a quasi-cyclic structure. It was incorporated into the fifth-generation mobile communication new wireless technology (5G NR) in 2018 to provide support for high-speed and large-capacity data transmission. Because it is close to the Shannon limit and has low coding complexity, it is often used in optical communications, satellite communications, information storage, fifth-generation mobile communications and other fields and plays an important role. QC-LDPC codes with fewer short loops usually have better performance, so it is of great significance to design QC-LDPC codes with large girth.

[0004] The construction method of QC-LDPC code can be divided into two categories: random construction method and algebraic construction method. However, the former is based on computer search to construct, and the structure of the code generator matrix and the check matrix has no definite form, so the constructed check matrix is ​​usually irregular, the encoding and decoding complexity is high, which is not conducive to hardware implementation, and is not widely used in practical applications, so scholars mostly adopt a construction method based on algebra. The present invention proposes a new construction method based on the algebraic construction method, first fixes the first two rows of sequences and the first element of the third row by algebraic calculation of generalized arithmetic progression, and then proposes a search algorithm based on generalized arithmetic progression through the four-ring and six-ring characteristics of the position of the element in the exponential matrix, eliminates the four-ring and six-ring in the check matrix, and finally designs a class of regular QC-LDPC codes with a row weight of L, a column weight of J, and a girth of at least 8, which has a simple construction method, low coding complexity, easy hardware implementation, good error correction performance and no obvious error flattening phenomenon. Summary of the invention

[0005] In view of this, the purpose of the present invention is to fix the first two rows of sequences and the first element of the third row based on the algebraic calculation of generalized arithmetic progression, and then use the search algorithm based on generalized arithmetic progression to obtain other elements of the third row, so as to form an exponential matrix (EM) that satisfies the condition of no 4 or 6 rings, and finally expand the exponential matrix accordingly to obtain its parity check matrix, and finally obtain a parity check matrix without four rings and six rings.

[0006] To achieve the above object, the present invention provides the following technical methods:

[0007] First, according to the target code length and the number of columns of the exponential matrix, the dimension of the circulant permutation matrix, that is, the expansion factor P, is determined.

[0008] Then determine the three elements b1, c1, d1 of the first column of the exponential matrix E, where b1<c1<d1, and then determine the arithmetic sequence first terms m1 and m2 and the generalized differences r1 and r2 that construct the first and second row sequences, where the arithmetic sequence first terms and the generalized differences satisfy m1<m2, r1<r2.

[0009] By determining the first column element and the first term of the arithmetic sequence and the generalized difference, the set b = {b1, b2, ..., b L} and set c={c1,c2,…,c L} as the first two rows of the exponential matrix, where b1<b2<…<b L ,c1<c2<…<c L Then, based on the first element d1 of the third row, combined with the ring length theorem, a search algorithm is designed according to formula (1) to search for the first term m3 of the arithmetic sequence and the generalized difference r3 to construct the set d = {d1, d1, …, d L} as the third row sequence of the exponential matrix E, where m2<m3, r2<r3, d1<d2<…<d L , due to the special properties of generalized arithmetic progression, the last element d in the third row L is the largest element in the exponential matrix, when P ≥ d L When there is no four-ring between the first row and the second and third rows (such as Figure 2 There is no four-ring between the second and third rows, and there is no first-class six-ring between the first, second and third rows (as shown in Figure 2 ), combined with the second type of six rings (such as Figure 2 The exponential matrix E (as shown in formula (3)) obtained by the ring length theorem does not have four or six rings. The specific steps of its search algorithm are as follows:

[0010] ① According to the size of the exponential matrix, determine the size of the expansion factor P and the first column elements b1, c1, d1, and construct the first terms m1 and m2 of the arithmetic sequence of the first two rows and the generalized differences r1 and r2.

[0011] ② Based on the first column elements and the first term of the arithmetic sequence and the generalized difference determined in ①, set b and set c are calculated using formula (1).

[0012] ③ In the search interval, the search algorithm is used to obtain m3 and r3 in formula (1) to determine the set d and obtain the exponential matrix E; it is only necessary to ensure that there is no second-type six-ring in the exponential matrix E. If there is a second-type six-ring, continue searching; if there is no second-type six-ring, determine the set d and then obtain the exponential matrix E.

[0013] Finally, the 0 elements in the exponential matrix E are replaced by the identity matrix of size P×P, and the non-zero elements are the shift coefficients of the circulant permutation matrix, that is, the identity matrix of size P×P is circulantly shifted to the right by the corresponding number of bits, and after expansion, a parity check matrix H of size 3P×LP is obtained.

[0014] Finally, through the above QC-LDPC code construction method, a class of regular QC-LDPC codes with a code length of LP and a code rate of (L-3) / L is constructed.

[0015] The beneficial effects of the present invention are:

[0016] The present invention relates to an 8-ring QC-LDPC code construction method based on generalized arithmetic progression. Further analysis of the construction steps of the construction method of the present invention shows that the code length of the codeword constructed by the method is only related to the dimension of the circulant permutation matrix (CPM), the code length and code rate are flexible and variable, the constructed check matrix has at least no four rings and six rings, that is, the ring length is at least 8, and has excellent error correction performance and no obvious error flattening phenomenon. Simulation results show that the GASS-QC-LDPC (2400, 1200) code constructed by this method is comparable to the regular SS-QC-LDPC (2400, 1200) code constructed based on Stanley sequence combined with search algorithm in reference [1] "Yuan Jianguo, Kuai Jiasong, Zhang Shuaikang. A novel construction method of QC-LDPC codes based on Stanley sequence [J]. Journal of Chongqing University of Posts and Telecommunications (Natural Science Edition), 2023, 35(3): 427-433.", the SD-QC-LDPC (2400, 1200) code constructed by algebraic construction based on Sidon sequence in reference [2] "L. Zhang and J. Wang, Construction of QC-LDPC Codes from Sidon Sequence Using Permutation and Segmentation, in IEEE Communications Letters, 2022, 26(08): 1710-1714.", and the reference [3] "ZHANG Guohua, NI Mengdi, HU Yulin, et al. Quasi-Cyclic LDPC Codes With Girth at Least Eight Based on Disjoint Difference Sets [J]. IEEE Communications Letters, 2023, 27 (1): 55-59. "Compared with the regular DDS-QC-LDPC (2400, 1200) code constructed based on disjoint difference sets (DDS), the net coding gain is improved to a certain extent. In summary, the codeword length obtained by the construction method provided by the present invention is only related to the dimension of the cyclic permutation matrix, and there are no four or six rings in the Tanner graph of the parity check matrix, which reduces the degree to which the code length is restricted by the sequence, makes it easier to construct a codeword with a shorter code length, and can better meet the requirements of the communication system. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] In order to make the purpose, technical method and beneficial effects of the present invention clearer, the present invention provides the following drawings for illustration:

[0018] Figure 1 A technical roadmap for the method of the present invention;

[0019] Figure 2 It is the type graph of 4-ring and 6-ring existence in the Tanner graph of the QC-LDPC code parity check matrix;

[0020] Figure 3 Performance comparison diagram of the QC-LDPC code constructed by the present invention and other code types when the code rate is 0.5; DETAILED DESCRIPTION

[0021] The preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0022] 1. Combine with attachment Figure 1 It is explained that a method for constructing an 8-ring QC-LDPC code based on a generalized arithmetic progression is as follows:

[0023] Step 1: According to the code length and code rate to be constructed, first determine the size of the expansion factor P and the three elements b1, c1, d1 of the first column of the exponential matrix E, where b1<c1<d1, and then determine the first terms m1 and m2 of the arithmetic sequence and the generalized difference r1 and r2 for constructing the first and second row sequences, where the first term of the arithmetic sequence and the generalized difference satisfy m1<m2, r1<r2.

[0024] Step 2: Based on the first column elements and the first term of the arithmetic sequence determined in step 1 and the generalized difference, the set b = {b1, b2, …, b L} and set c={c1,c2,…,c L} as the first two rows of the exponential matrix, where b1<b2<...<b L ,c1<c2<...<c L .

[0025]

[0026] Where t represents the number of rows, o represents the number of columns, and n t is the first term of the generalized arithmetic sequence, m t is the first term of the arithmetic sequence, r t is the generalized difference, where n t , m t and r t It increases with the increase of t;

[0027] Step 3: Based on the first element d1 of the third row determined in step 1, combined with the ring length theorem, a search algorithm is designed according to formula (1) to search for the first term m3 of the arithmetic sequence and the generalized difference r3 to construct a set d = {d1, d1, …, d L} as the third row sequence of the exponential matrix E, where m2<m3, r2<r3, d1<d2<…<d L , due to the special properties of generalized arithmetic progression, the last element d in the third row L is the largest element in the exponential matrix, when P ≥ d L When there is no four-ring between the first row and the second and third rows (such as Figure 2 There is no four-ring between the second and third rows, and there is no first-class six-ring between the first, second and third rows (as shown in Figure 2 ), combined with the second type of six rings (such as Figure 2 The exponential matrix E (as shown in formula (3)) obtained by the ring length theorem does not have four or six rings;

[0028] Ring length theorem: In QC-LDPC (J, L) codes, the necessary and sufficient condition for the existence of a 2n ring is (2).

[0029]

[0030] (2) Where: j0 = j L ,i l ≠i l+1 ,j l ≠j l+1 ,0≤i l <J,0≤j l <L, where 0≤l≤n-1;

[0031]

[0032] In order to ensure that there are no four or six rings in the Tanner graph corresponding to the constructed QC-LDPC code parity check matrix, the search algorithm is designed by combining the ring length theorem with the four or six ring characteristics of the position of the elements in the exponential matrix. The specific steps of the search algorithm are as follows:

[0033] ① According to the size of the exponential matrix, determine the size of the expansion factor P and the first column elements b1, c1, d1, and construct the first terms m1 and m2 of the arithmetic sequence of the first two rows and the generalized differences r1 and r2.

[0034] ② Based on the first column elements and the first term of the arithmetic sequence and the generalized difference determined in ①, set b and set c are calculated using formula (1).

[0035] ③ In the search interval, the search algorithm is used to obtain m3 and r3 in formula (1) to determine the set d and obtain the exponential matrix E; it is only necessary to ensure that there is no second-type six-ring in the exponential matrix E. If there is a second-type six-ring, continue searching; if there is no second-type six-ring, determine the set d and then obtain the exponential matrix E.

[0036] Step 4: Replace the 0 elements in the exponential matrix E with the identity matrix of size P×P. The non-zero elements are the shift coefficients of the circulant permutation matrix, that is, the identity matrix of size P×P is circulantly shifted to the right by the corresponding number of bits. After expansion, a parity check matrix H of size 3P×LP is obtained.

[0037] 2. Combine with attachment Figure 2 It is shown that there are no four-rings or six-rings in the Tanner graph corresponding to the QC-LDPC code parity check matrix constructed by the present invention.

[0038] (a) Proof that there are no four rings in the exponential matrix.

[0039] Attached Figure 2 The structure of the existence of four rings in the Tanner graph is given in the paper. According to the characteristics of generalized arithmetic progression, when P ≥ d L , it is proved that there is no tetracycle in the exponential matrix.

[0040] Proof: The elements in the exponential matrix need to satisfy equation (4) when there is a four-ring

[0041]

[0042] Among them, 1≤j0<j1≤3, 1≤l0<l1≤L.

[0043] Substituting formula (1) into formula (4) yields formula (5) as follows:

[0044]

[0045] Where m1≤m i <m s ≤m3,r1≤r i <r s ≤r3

[0046] By scaling formula (5), we can use replace The formula (6) is as follows:

[0047]

[0048] Take the maximum value of the exponential matrix E as p 3,L =d L , when P ≥ d L When , the exponential matrix has no quadruple rings.

[0049] (ii) There is no first-class six-ring proof in the exponential matrix.

[0050] Attached Figure 2 The structure of the first type of six rings in the Tanner graph is given in the paper. According to the characteristics of generalized arithmetic progression, when P ≥ d LWhen , it is proved that there is no hexacyclic ring of the first kind in the exponential matrix.

[0051] The first kind of six-ring proof: the existence of exponential matrix Figure 2 The six rings of type (a) need to satisfy formula (7)

[0052]

[0053] Among them, 1≤j0<j1<j2≤3, 1≤l0<l1<l2≤L.

[0054] Substituting formula (1) into formula (7), we get formula (8) as follows:

[0055]

[0056] By scaling formula (8), we can use replace The formula (9) is as follows:

[0057]

[0058] By scaling formula (9), we can use replace We get formula (10):

[0059]

[0060] From the above, we can see that the maximum value of the exponential matrix E is p 3,L =d L , when P ≥ d L When , the exponential matrix does not have Figure 2 Medium (a) class six rings.

[0061] Similarly, Figure 2 The six-ring structures of type (b), (c), and (d) are obtained by calculation and scaling to obtain formula (11):

[0062]

[0063] When P ≥ d L When , there is no such thing in the exponential matrix Figure 2 The six rings in.

[0064] In summary, according to the special properties of generalized arithmetic progression, when the first element of the first row, the second row and the third row satisfies b1<c1<d1, and the first term of the arithmetic progression formed by the difference of each row of generalized arithmetic progression is m1<m2<m3 and the generalized difference r1<r2<r3, P≥d L When , there are no four-rings and first-type six-rings in the constructed exponential matrix E.

[0065] (II) Proof that exponential matrices do not have a second-type six-ring

[0066] Attached Figure 2 The structural diagram of the existence of quadruple rings and hexacycles in the Tanner graph is given. By analyzing the construction method, it can be seen that it is impossible for an exponential matrix to have quadruple rings and first-class hexacycles, and the search algorithm obviously avoids the generation of quadruple rings and hexacycles in the entire exponential matrix. Therefore, this construction method can construct a class of QC-LDPC codes with a column weight of 3, a code rate of (L-3) / L, and a girth of 8.

[0067] 3. Combine with attachment Figure 3 Note: In order to verify that the constructed codeword has better error correction performance, MATLAB is used for simulation. Considering the row weight L = 6, column weight J = 3, the first column element is fixed to b1 = 7, c1 = 15, d1 = 25, the first term m1 and m2 of the first and second rows of the arithmetic sequence are constructed to be 3 and 9 respectively, and their generalized differences r1 and r2 are 1 and 3 respectively, and the sequences b = {7, 10, 14, 19, 25, 32} and c = {15, 24, 36, 51, 69, 90} are obtained. The expansion factor P value is 400, and the search interval of the first term m3 of the arithmetic sequence of the third row is [10, 15], and the search interval of its generalized difference r3 is [6, 15], and m3 = 10, r3 = 13 are obtained, so the sequence d = {25, 35, 58, 94, 143, 205} is obtained as the third row of the exponential matrix. The exponential matrix E is obtained as shown in formula (12):

[0068]

[0069] The error correction performance of the constructed GR-QC-LDPC code is verified by simulation using MATLAB software. The channel adopts Additive White Gaussian Noise (AWGN) channel, Binary Phase Shift Keying (BPSK) modulation, and Belief Propagation (BP) algorithm is selected for decoding. The maximum number of decoding iterations is 50. The constructed GASS-QC-LDPC (2400, 1200) code is compared with the regular SS-QC-LDPC (2400, 1200) code constructed based on Stanley sequence combined with search algorithm in the literature [1] "Yuan Jianguo, Kuai Jiasong, Zhang Shuaikang. A novel construction method of QC-LDPC code based on Stanley sequence [J]. Journal of Chongqing University of Posts and Telecommunications (Natural Science Edition), 2023, 35(3): 427-433.", and the literature [2] "L. Zhang and J. Wang, Construction of QC-LDPC Codes from Sidon Sequence Using Permutation and Segmentation, in IEEE Communications Letters, 2022, 26(08):1710-1714.” and the regular DDS-QC-LDPC (2400, 1200) code constructed based on disjoint difference sets (DDS) in the literature [3] “ZHANG Guohua, NI Mengdi, HU Yulin, et al. Quasi-Cyclic LDPC Codes With Girth at Least Eight Based on Disjoint Difference Sets[J]. IEEE Communications Letters, 2023, 27(1):55-59.”. The simulation curve comparison between the bit error rate (BER) and signal-to-noise ratio (SNR) of different QC-LDPC codes under the same parameters is shown in the attached figure. Figure 3 As shown. Figure 3 It can be seen that when the BER is 10 -6The net coding gain (NCG) of the GASS-QC-LDPC (2400, 1200) code constructed by the present invention is improved by about 0.2dB, 0.51dB and 0.91dB respectively compared with the SS-QC-LDPC (2400, 11200) code, SD-QC-LDPC (2400, 1200) code and DDS-QC-LDPC (2400, 1200) code.

[0070] Based on the above simulation verification, it can be known that the present invention proposes a regular QC-LDPC code construction method using generalized arithmetic progression combined with a search algorithm. The parity check matrix Tanner diagram corresponding to the codeword does not contain four or six rings. The simulation results show that under the same conditions, the net coding gain NCG of the GASS-QC-LDPC code constructed by the present invention is improved to a certain extent compared with the SD-QC-LDPC code, DDS-QC-LDPC code and SS-QC-LDPC code with the same code rate and code length, and the error correction performance is good without obvious error flattening phenomenon.

[0071] Finally, it should be noted that the above preferred embodiments are only used to illustrate the technical solutions of the present invention rather than to limit it. Although the present invention has been described in detail through the above preferred embodiments, those skilled in the art should understand that various changes can be made in form and details without departing from the scope defined by the claims of the present invention.

Claims

1. A method for constructing an 8-ring QC-LDPC (Quasi-Cyclic Low-Density Parity-Check, QC-LDPC) code based on a generalized arithmetic progression. The method first fixes the first two rows of sequences and the first element of the third row by algebraic calculation of the generalized arithmetic progression, then uses a search algorithm based on the generalized arithmetic progression to obtain other elements of the third row, and constructs an exponent matrix (exponent matrix, EM) that satisfies the condition of no 4 or 6 rings. Finally, the exponent matrix is ​​expanded to obtain its parity check matrix, so that there are no 4 or 6 rings in the Tanner graph of the check matrix, and the required storage space is small, and it is easy to implement in hardware; The specific steps are as follows: Step 1: According to the code length and code rate to be constructed, first determine the size of the expansion factor P and the three elements b1, c1, d1 of the first column of the exponential matrix E, where b1<c1<d1, and then determine the first term m1 and m2 of the arithmetic sequence and the generalized difference r1 and r2 for constructing the first and second row sequences, where the first term and the generalized difference satisfy m1<m2, r1<r2; Step 2: Based on the first column elements and the first term of the arithmetic sequence determined in step 1 and the generalized difference, the set b = {b1, b2, …, b L } and set c={c1,c2,…,c L } as the first two rows of the exponential matrix, where b1<b2<…<b L ,c1<c2<…<c L ; Where t represents the number of rows, o represents the number of columns, and n t is the first term of the generalized arithmetic sequence, m t is the first term of the arithmetic sequence, r t is the generalized difference, where n t , m t and r t It increases with the increase of t; Step 3: Based on the first element d1 of the third row determined in step 1, combined with the ring length theorem, a search algorithm is designed according to formula (1) to search for the first term m3 of the arithmetic sequence and the generalized difference r3 to construct the set d = {d1, d1,, d L } as the third row sequence of the exponential matrix E, where m2<m3, r2<r3, d1<d2<…<d L , due to the special properties of generalized arithmetic progression, the last element d in the third row L is the largest element in the exponential matrix, when P ≥ d L When , there is no four-ring between the first row and the second row and the third row, and there is no four-ring between the second row and the third row. At the same time, there is no first-type six-ring between the first row, the second row and the third row. In the algorithm search, the exponential matrix E (as shown in formula (3)) obtained by combining the ring length theorem of the second-type six-ring does not have a four-six ring; Ring length theorem: In QC-LDPC (J, L) codes, the necessary and sufficient condition for the existence of a 2n ring is (2). (2) Where: j0 = j L ,i l ≠i l+1 ,j l ≠j l+1 ,0≤i l <J,0≤j l <L, where 0≤l≤n-1; In order to ensure that there are no four or six rings in the Tanner graph corresponding to the constructed QC-LDPC code parity check matrix, the search algorithm is designed by combining the ring length theorem with the four or six ring characteristics of the position of the elements in the exponential matrix. The specific steps of the search algorithm are as follows: ① According to the size of the exponential matrix, determine the size of the expansion factor P and the first column elements b1, c1, d1, and construct the first terms m1 and m2 of the arithmetic sequence of the first two rows and the generalized difference values ​​r1 and r2; ② According to the first column elements and the first term of the arithmetic sequence and the generalized difference determined in ①, set b and set c are calculated by formula (1); ③ In the search interval, the m3 and r3 in formula (1) are obtained by the search algorithm to determine the set d and obtain the exponential matrix E; it is only necessary to ensure that there is no second-class six-ring in the exponential matrix E. If there is a second-class six-ring, continue searching; if there is no second-class six-ring, determine the set d and then obtain the exponential matrix E; Step 4: Replace the 0 elements in the exponential matrix E with the unit matrix of size P×P. The non-zero elements are the shift coefficients of the circulant permutation matrix, that is, the unit matrix of size P×P is circulantly shifted to the right by the corresponding number of bits. After expansion, a parity check matrix H of size 3P×LP is obtained.

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