Eight-ring QC-LDPC code construction method based on subtraction construction

Abstract

The invention relates to an eight-ring QC-LDPC (Quasi-Cyclic Low Density Parity Check) code construction method based on a subtraction construction search algorithm. According to the method, a first row is fixed by using a Golomb Ruler sequence, the last two rows are obtained by using a search algorithm constructed based on subtraction to form an index matrix meeting a condition without 4 and 6 rings, 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, a GR-QC-LDPC code, an LETS-QC-LDPC code and a GRS-QC-LDPC code which have the same code rate and code length, the SC-QC-LDPC code constructed by the method has the advantages that the net coding gain is improved to a certain degree, and the constructed SC-QC-LDPC code has better waterfall region performance in a high signal-to-noise ratio region and has no 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 subtraction construction. 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 included in the fifth-generation mobile communication new wireless technology (5GNR) 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 algebraic construction methods. The present invention proposes a new construction method based on the algebraic construction method, first determines the target code length to be constructed, then selects the elements in the Golomb Ruler, and then proposes a search algorithm based on the subtraction construction through the four-ring and six-ring characteristics of the position of the elements in the exponential matrix, eliminates the four-ring and six-ring in the check matrix, and finally designs a regular QC-LDPC code 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 object of the present invention is to provide an 8-ring QC-LDPC code construction method based on subtraction construction. The method first uses a Golomb Ruler sequence to fix the first row, and the next two rows are obtained by a search algorithm based on subtraction construction, which can form an exponent matrix (EM) that satisfies the condition of no 4 or 6 rings, and finally the exponent matrix is ​​correspondingly expanded to obtain its parity check matrix, and finally a parity check matrix without four or six rings is obtained.

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

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

[0008] Then, construct a set a as the first row of the exponential matrix. Select a Golomb Ruler of order L to construct a set a = {a0, a1, ..., a L-1}, where a0 <a1<…<a L-1 Combining the ring length theorem and the characteristics of four and six rings, a search algorithm is designed based on the subtraction construction to construct the sequence. The search method is used to search for the set b={b0,b1,…,b L-1} As the second line, the set c = {c0, c1, ..., c L-1} as the third line, where b0 <b1<…<b L-1 ,c0 <c1<…<c L-1 , due to the special properties of the subtraction construction, when P>(L-1)max{e 1,1 ,e 1,2 ,…,e 1,L-1}, there is no quadruple ring between the second and third rows. Since the first row is fixed as the Golomb Ruler sequence, the exponential matrix E obtained by combining the ring length theorem in the algorithm search does not have quadruple or hexacyclic rings. The specific steps of the search algorithm are as follows:

[0009] ① Given the size of P, select the order and number of the Golomb Ruler according to the size of the exponential matrix and fix it to the first row of the exponential matrix. Set a suitable search interval. In order to reduce the search complexity, set the search interval to [0,30].

[0010] ② In the search interval, the number of set b is first determined by the search algorithm. After the number of set b is determined, the number of set c is calculated by formula (1) to form the exponential matrix E. Due to the special properties of the subtraction construction, there is no quadricycle in the second and third rows. It is only necessary to ensure that there is no quadricycle between the second and third rows and the first row. If there is, continue searching. If not, determine set b.

[0011] ③ Use the search algorithm to detect whether the exponential matrix obtained in step 2 satisfies Theorem 1 when n=3, that is, whether there are six rings. If there are six rings, repeat step 2. If not, determine set b and obtain the exponential matrix E.

[0012] 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.

[0013] 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.

[0014] The beneficial effects of the present invention are:

[0015] The present invention relates to a method for constructing an 8-ring QC-LDPC code based on subtraction construction. 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, and the constructed check matrix has at least no four-ring and six-ring, 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 SC-QC-LDPC (1200, 600) code constructed by this method is comparable to the SS-QC-LDPC (1200, 600) code with the same code length and code rate as the SS-QC-LDPC (1200, 600) code constructed based on Sidon sequence algebraically in reference [1] "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 SS-QC-LDPC (1200, 600) code constructed based on Sidon sequence algebraically in reference [2] "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 (1200, 600) code constructed by using the original prototype graph to eliminate the leafless basic trap set in the literature [3] "KARIMI B, BANIHASHEMI A H. Construction of Irregular Protograph-Based QC-LDPC Codes With Low Error Floor[J]. IEEE Transactions on Communications, 2021, 69(1):3-18.", the net coding gain has been 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 circulant 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 codewords with shorter code lengths, and can better meet the requirements of the communication system. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] 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:

[0017] Figure 1 A technical roadmap for the method of the present invention;

[0018] 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;

[0019] 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

[0020] The preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0021] 1. Combine with attachment Figure 1 It is explained that a method for constructing an 8-ring QC-LDPC code based on subtraction construction is as follows:

[0022] Step 1: Determine the size of the cyclic shift matrix, that is, the expansion factor P, according to the code length to be constructed and the number of columns of the exponential matrix.

[0023] Step 2: Select a Golomb Ruler sequence of order L and construct a set a = {a0, a1, …, a L-1}, where a0 <a1<…<a L-1 .

[0024] Step 3: Combining the ring length theorem and the characteristics of four and six rings, design a search algorithm to construct a sequence according to formula (1). Use the search method to search for the set b = {b0, b1, ..., b L-1} As the second line, the set c = {c0, c1, ..., c L-1} as the third line, where b0 <b1<…<b L-1 ,c0 <c1<…<c L-1 , due to the special properties of the subtraction construction, when P>(L-1)max{e 1,1 ,e 1,2 ,…,e 1,L-1}, Theorem 1 is not satisfied between the second and third rows, so there is no quadruple ring in these two rows. Since the first row is fixed as the Golomb Ruler sequence, the exponential matrix E obtained by combining the ring length theorem in the algorithm search is shown in Formula (3), and there is no quadruple or hexacyclic ring.

[0025]

[0026] where 1 < t < L - 1.

[0027] Theorem 1: In the QC-LDPC(J,L) code, the necessary and sufficient condition for the existence of a 2n-cycle is Equation (2);

[0028]

[0029] In Equation (2): 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.

[0030]

[0031] To ensure that there are no 4-cycles and 6-cycles in the Tanner graph corresponding to the parity-check matrix of the constructed QC-LDPC code, the ring length theorem and the 4-cycle and 6-cycle characteristics of the positions of the elements in the exponent matrix are combined to design its search algorithm. The specific steps of this search algorithm are as follows:

[0032] ① Given the size of P, select the order and number of the Golomb Ruler according to the size of the exponent matrix, and fix it as the first row of the exponent matrix. Set a suitable search interval. To reduce the search complexity, set the search interval as [0, 30].

[0033] ② In the search interval, first determine the numbers in set b through the search algorithm. After determining the numbers in set b, calculate the numbers in set c through Equation (1) to form the exponent matrix E. Due to the special properties of subtraction construction, there are no 4-cycles in the second and third rows. It is only necessary to ensure that there are no 4-cycles between the second and third rows and the first row. If there are, continue the search; if not, determine set b.

[0034] ③ Use the search algorithm to detect whether the exponent matrix obtained in step 2 satisfies Theorem 1 when n = 3, that is, whether there are 6-cycles. If there are 6-cycles, repeat step 2; if not, determine set b, and then obtain the exponent matrix E.

[0035] Step 4: Replace the 0 elements in the exponent matrix E with the identity matrix of size P×P, and the non-0 elements are the shift coefficients of the cyclic permutation matrix, that is, shift the identity matrix of size P×P to the right by the corresponding number of bits, and expand it to obtain the parity-check matrix H of size 3P×LP.

[0036] 2. Combining the appendix Figure 2 It shows that there are no 4-cycles and 6-cycles in the Tanner graph corresponding to the parity-check matrix of the QC-LDPC code constructed by the present invention.

[0037] (1) Proof of no four - cycle between the second and third rows of the exponent matrix.

[0038] Appendix Figure 2 The structure of the four - cycle in the Tanner graph is given. According to the cycle - length theorem and the characteristics of subtraction construction, when P > (L - 1)max{e 1,1 , e 1,2 , …, e 1,L-1}, it can be proved that there is no four - cycle between the first and second rows.

[0039] Proof: There is a four - cycle when the second and third rows satisfy Equation (4)

[0040] e 1,i -e 1,j +e 2,j -e 2,i ≡0 (mod P)(4) where j > i.

[0041] It can be seen from Equation (1) that the elements of the third row are obtained from the second row. Therefore, substituting Equation (1) into Equation (4) can obtain Equation (5) as follows:

[0042] When 0 < i < j < L - 1

[0043] e 1,i -e 1,j +e 2,j-1 +e 1,j +(max{e 1,1 , e 1,2 , …, e 1,L-1}-e 1,j )

[0044] -e 2,i-1 +e 1,i +(max{e 1,1 , e 1,2 , …, e 1,L-1}-e 1,i )≡0 (mod P)

[0045] (5) After simplification, Equation (6) is obtained as follows:

[0046] e 2,j-1 -e 2,i-1 ≡0 (mod P)(6) Due to the special property of subtraction construction, the second and third rows are two increasing sequences greater than 0, and Equation (7) can be deduced

[0047] e 2,j-1 -e 2,i-1 ≠0 (7)

[0048] When 0 < i < j = L = 1

[0049] Repeat the above steps and substitute into formula (1) to obtain formula (8)

[0050] e 1,i -e 1,L-1 +e 2,L-2 +e 1,L-1 +(max{e 1,1 ,e 1,2 ,…,e 1,L-1}-e 1,1}

[0051] -e 2,i-1 -e 1,i -(max{e 1,1 ,e 1,2 ,…,e 1,L-1}-e 1,i}≡0(modP)

[0052] (8) is simplified to obtain formula (9) as follows:

[0053] e 2,L-2 -e 1,1 +e 1,i -e 2,i-1 ≡0(modP)(9) Because of the special properties of the subtraction construction, the second and third rows are two increasing sequences greater than 0, and equation (10) can be derived

[0054] e 2,L-2 -e 1,1 +e 1,i -e 2,i-1 ≠0 (10)

[0055] Therefore, in summary, there is no four-ring between the second and third rows.

[0056] By mathematical induction, we can get that when P>(L-1)max{e 1,1 ,e 1,2 ,…,e 1,L-1}, there are no quad rings in the second and third rows.

[0057] (II) Proof that exponential matrices do not have four or six rings

[0058] 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 there can be no quadruple ring between the second and third rows of the exponential matrix, 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.

[0059] 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. Consider constructing a class of QC-LDPC codes with row weight J = 6 and column weight L = 3. The Golomb Ruler of order 6 is used as set a, a = {0, 1, 4, 11, 26, 32}. The search interval of the first two numbers in the sequence is set to [1, 10], and the search interval of the last three numbers is set to [11, 40], and the sequences b = {0, 9, 10, 13, 22, 35} and c = {0, 44, 79, 114, 149, 210} are obtained. When P = 200, using the search algorithm proposed in this paper, an exponential matrix can be constructed, and the structure is shown in formula (11):

[0060]

[0061] 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, and the SC-QC-LDPC (1200, 600) code is compared with the SS-QC-LDPC (1200, 600) code with the same code length and code rate in the literature [1] "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 SS-QC-LDPC (1200, 600) code based on the algebraic construction of Sidon sequence in the literature [2] "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 performance of the regular DDS-QC-LDPC (1200, 600) code constructed based on the original prototype graph and eliminating the leafless basic trap sets in the literature [3] "KARIMI B, BANIHASHEMI A H. Construction of Irregular Protograph-Based QC-LDPC Codes With Low Error Floor[J]. IEEE Transactions on Communications, 2021, 69(1):3-18." is compared.

[0062] The comparison of 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 3It can be seen that when the BER is 10 -6 The net coding gain (NCG) of the GR-QC-LDPC (1200, 600) code constructed by the present invention is improved by about 0.23dB, 0.39dB and 0.74dB respectively compared with the DDS-QC-LDPC (1200, 600) code, LETS-QC-LDPC (1200, 600) code and MT-QC-LDPC (1200, 600) code.

[0063] Based on the above simulation verification, it can be known that the present invention proposes a regular QC-LDPC code construction method using subtraction construction 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 SC-QC-LDPC code constructed by the present invention has a certain improvement in net coding gain NCG compared with the LETS-QC-LDPC code, DDS-QC-LDPC code and SS-QC-LDPC code with the same code rate and code length, and has good error correction performance and no obvious error flattening phenomenon.

[0064] 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 subtraction construction search algorithm, wherein the method first uses a Golomb Ruler sequence to fix the first row, and the next two rows are obtained by a subtraction-based search algorithm to form an exponent matrix (EM) that satisfies the condition of no 4 or 6 rings, and 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: Determine the size of the cyclic shift matrix, i.e., the expansion factor P, according to the code length to be constructed and the number of columns of the exponential matrix; Step 2: Select a Golomb Ruler sequence of order L and construct a set a = {a0, a1, …, a L-1 }, where a0 <a1<…<a L-1 ; Step 3: Combining the ring length theorem and the characteristics of four and six rings, design a search algorithm to construct a sequence according to formula (1). Use the search algorithm to search for the set b = {b0, b1, ..., b L-1 } As the second line, the set c = {c0, c1, ..., c L-1 } as the third line, where b0 <b1<…<b L-1 ,c0 <c1<…<c L-1 , due to the special properties of the subtraction construction, when P>(L-1)max{e 1,1 ,e 1,2 ,…,e 1,L-1 }, the second and third rows do not satisfy Theorem 1, so there are no four-rings in these two rows. Since the first row is fixed as the Golomb Ruler sequence, the exponential matrix E obtained by combining the ring length theorem in the algorithm search is shown in formula (3), and there are no four or six rings; 1 of them <t<L-1; Theorem 1: In QC-LDPC (J, L) codes, the necessary and sufficient condition for the existence of a 2n ring is (2). In formula (2): 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 Given the size of P, select the order and number of the Golomb Ruler according to the size of the exponential matrix, fix it to the first row of the exponential matrix, and set a suitable search interval. In order to reduce the search complexity, set the search interval to [0,30]; ② In the search interval, the number of set b is first determined by the search algorithm. After the number of set b is determined, the number of set c is calculated by formula (1) to form the exponential matrix E. Due to the special properties of the subtraction construction, there is no quadricycle in the second and third rows. It is only necessary to ensure that there is no quadricycle between the second and third rows and the first row. If there is, continue searching. If not, determine set b. ③ Use the search algorithm to detect whether the exponential matrix obtained in step 2 satisfies Theorem 1 when n=3, that is, whether there are six rings. If there are six rings, repeat step 2. If not, determine the set b, and then obtain the exponential matrix E. 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.

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