Short code length QC-LDPC code construction method based on Stanley sequence and genetic algorithm

Abstract

The invention relates to a short code length QC-LDPC (Quasi-Cyclic Low-Density Parity-Check) code construction method based on a Stanley sequence and a genetic algorithm. According to the method, a Stanley sequence is used for forming first two rows of an index matrix meeting the condition that no four rings exist, then a genetic algorithm is used for searching a third row element which enables the index matrix to meet the requirements that the ring length is at least 8 and the number of trap sets is small, and therefore the index matrix with the girth length being at least 8 is obtained, the genetic algorithm can guarantee that the number of the trap sets is minimum, and the number of trap sets is small. And finally, the index matrix is 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 number of trap sets is small, the storage space required by code words is small, and hardware implementation is easy. Simulation shows that compared with an HS-QC-LDPC code, an MT-QC-LDPC code and an SES-QC-LDPC code which have the same code rate and code length, the SS-QC-LDPC code constructed by the method has the advantages that the net coding gain is improved to a certain extent, and the constructed SS-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] This invention belongs to the field of channel coding technology and relates to a method for constructing short code-length quasi-cyclic low-density parity-check (QC-LDPC) codes based on Stanley sequences and genetic algorithms. Background Technology

[0002] The basic purpose of communication is to effectively and reliably transmit information generated by a source to a destination. However, during information transmission, various kinds of interference are inevitable, which can lead to bit errors in communication and reduce the effectiveness and reliability of information transmission. Therefore, channel coding technology is needed to ensure reliable information transmission.

[0003] QC-LDPC codes are a classic class of linear block error-correcting codes. Their parity-check matrix (PCM) possesses quasi-cyclic properties, and in 2018, they officially became the core coding scheme for 5G NR (5G New Radio) technology, providing reliable support for high-speed, high-capacity data transmission. Due to their error-correcting performance approaching the Shannon limit and low encoding / decoding complexity, this type of coding has been widely used in several key areas such as optical communication, satellite communication, information storage, and 5G. Short-code-length QC-LDPC codes offer significant advantages, including low latency, ease of implementation, and adaptability to real-time application scenarios. Therefore, designing high-performance short-code-length QC-LDPC codes is of paramount importance for the physical layer design of high-real-time, high-reliability communications.

[0004] The construction methods of QC-LDPC codes can be divided into two main categories: random construction methods and algebraic construction methods. The former is based on computer search, and the structure of the code generator matrix and parity check matrix has no fixed form. Therefore, the constructed parity check matrix is ​​usually irregular, resulting in high encoding and decoding complexity, which is not conducive to hardware implementation and is not widely used in practical applications. Therefore, researchers mostly adopt algebraic construction methods. This invention proposes a new construction method based on algebraic construction. First, the first two rows of an exponent matrix satisfying the condition of no four-rings are constructed using Stanley sequences. Then, a genetic algorithm is used to search for the third row elements that satisfy the condition of at least 8 rings and a small number of trap sets, constructing the corresponding exponent matrix and obtaining its parity check matrix. This ensures that the Tanner graph of the parity check matrix does not contain four-rings or six-rings and has only a small number of trap sets. Finally, a short-length QC-LDPC code with row weight L, column weight J, and a ring length of at least 8 is designed. Its construction method is simple, the encoding complexity is low, it is easy to implement in hardware, the error correction performance is good, and there is no obvious error layering phenomenon. Summary of the Invention

[0005] In view of this, the purpose of this invention is to provide a method for constructing short code-length QC-LDPC codes based on Stanley sequences and genetic algorithms. This method first uses Stanley sequences to construct the first two rows of an exponent matrix that satisfies the condition of no 4-rings. Then, it uses a genetic algorithm to construct an exponent matrix that satisfies the condition of a ring length of at least 8 and a small number of trap sets. The exponent matrix is ​​then expanded to finally obtain the required parity check matrix.

[0006] To achieve the above objectives, the present invention provides the following technical method:

[0007] First, construct the first two rows of the QC-LDPC(3,L) code exponent matrix A(H) that satisfies the 4-ring-free condition using the Stanley sequence. The elements of the first and second rows are obtained using the Stanley sequence b. i ={b0,b1,…,b L-1 We obtain that b0 < b1 < ... < b L-1 The expression is A(i,j)=(i+1)b j The elements in the third row of the exponent matrix are represented by -1, i.e., A(2,j)=-1,0≤j≤L-1.

[0008] Then, combining Lemma 1, the ring length theorem, and the properties of trap sets, a genetic algorithm is designed to search for a set c. i ={c0,c1,…,c L-1 Given a set c where all elements are non-negative integers, set c... i The elements in the matrix are used to replace the third row of A(H) to obtain E(H), which makes the check matrix free of four- or six-ring loops and has a smaller set of traps.

[0009] The specific steps of the genetic algorithm are as follows:

[0010] ① Determine the size of the expansion factor q based on the required code length;

[0011] ② The search interval is [0, q]. Within this interval, sequences that do not satisfy the existence theorem of cycle length are searched as individuals in the population. The number of trap sets is introduced as the fitness. The fitness of each individual is calculated, and the optimal individual is recorded. Selection, crossover, mutation, and constraint repair operations are performed on the population to update the individuals, and the process is iterated to find the globally optimal solution c with the minimum number of trap sets. i ={c0,c1,…,c L-1};

[0012] ③ Use the sequence c obtained in step ② i Construct the exponential matrix E(H).

[0013] Finally, the zero elements in the exponent matrix are replaced with an identity matrix of size q×q, and the non-zero elements are the shift coefficients of the cyclic permutation matrix, that is, the identity matrix of size q×q is cyclically shifted to the right by the corresponding number of bits, and after expansion, a parity check matrix H of size 3q×Lq is obtained.

[0014] Finally, using the aforementioned QC-LDPC code construction method, a class of regular QC-LDPC codes with a code length of qL and a code rate of (L-3) / L was constructed. Under the same simulation environment, the short-code-length QC-LDPC code construction method based on Stanley sequences and genetic algorithms proposed in this patent was compared and analyzed with other similar coding schemes.

[0015] The beneficial effects of this invention are as follows:

[0016] This invention relates to a method for constructing short codewords in QC-LDPC codes based on Stanley sequences and genetic algorithms. Further analysis of the construction steps reveals that the codeword length constructed by this method is only related to the dimension of the circulant permutation matrix (CPM), allowing for flexible code length and rate adjustments. The constructed parity-check matrix has a ring length of at least 8, and the number of trap sets is relatively small. Therefore, it possesses excellent error correction performance and exhibits no significant error layering phenomenon. Simulation results show that the SS-QC-LDPC(415,166) code constructed by this method is comparable to the regular HS-QC-LDPC(415,166) code with a ring length of 8 constructed based on Hoey sequence (HS) in reference [1] "Yuan Jianguo, Song Wanchuang. A method for constructing 8-ring QC-LDPC codes based on Hoey sequence [J]. Telecommunications Technology, 2025, 65(05):793-799." and the modified Tanner (Modified) code constructed based on Hoey sequence (HS) in reference [2] "Kim W, Cho H, Song HY, Song MK. Some variations of Tanner's construction for short length QC-LDPC codes [J]. Electronics Letters. 2024, 60(3):e13088." Compared with the regular MT-QC-LDPC(415,166) code constructed by Tanner (MT), and the regular SES-QC-LDPC(413,177) code based on the improved SES algorithm in reference [3] "A. Ovinnikov, A. Kharin, and K. Zavertkin et al. Small Elementary Trapping Sets Elimination of Girth-8 QC-LDPC Codes[C]. Digital Signal Processing and its Applications, Moscow, Russian Federation, 2024:1-5.", the net coding gain is improved to a certain extent. Attached Figure Description

[0017] To make the objective, technical method, and beneficial effects of this invention clearer, the following figures are provided for illustration:

[0018] Figure 1 This is a technical roadmap of the method of the present invention;

[0019] Figure 2The Tanner diagram for the parity check matrix of QC-LDPC code shows the existence types of rings 4 and 6;

[0020] Figure 3 The graph shows the performance comparison between the short code length QC-LDPC code constructed in this invention and other code types when the code rate is 0.4. Detailed Implementation

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

[0022] 1. Combined with the appendix Figure 1 The following is a method for constructing short code-length QC-LDPC codes based on Stanley sequences and genetic algorithms:

[0023] Step 1: Construct the first two rows of the QC-LDPC(3,L) code exponent matrix A(H) that satisfies the 4-ring-free condition using the Stanley sequence, as shown in equation (1); the elements of the first and second rows are obtained by using the Stanley sequence b. i ={b0,b1,…,b L-1 We obtain that b0 < b1 < ... < b L-1 This can be expressed as A(i,j)=(i+1)b j , where 0≤i≤1, 0≤j≤L-1; the elements in the third row of the exponent matrix are represented by -1, i.e., A(2,j)=-1,0≤j≤L-1;

[0024]

[0025] Step 2: Combining Lemma 1, the Ring Length Theorem, and the properties of trap sets, design a genetic algorithm to search and obtain a set c. i ={c0,c1,…,c L-1}, where all elements in set c are non-negative integers; set c i The elements in A(H) are replaced to obtain E(H), so that there are no four- or six-ring loops in the check matrix and only a small number of trap sets. E(H) is shown in equation (3).

[0026] Lemma 1: In QC-LDPC(m,n) codes, the necessary and sufficient condition for the existence of a 2c ring is equation (2);

[0027]

[0028] (2) Where: j0=j n i l ≠i l+1 ,j l ≠j l+1 ,0≤i l <m, 0≤j l<n, where 0≤l≤c-1;

[0029]

[0030] Since there are no 4-rings between the second and third rows of A(H), this genetic algorithm only needs to ensure that there are no 4- or 6-rings between the third row and other rows while reducing the trap set;

[0031] The specific steps of the genetic algorithm are as follows:

[0032] ① Determine the size of the expansion factor q based on the required code length;

[0033] ② The search interval is [0, q]. Within this interval, sequences that do not satisfy the existence theorem of the ring length in equation (2) are searched as individuals in the population. The number of trap sets is introduced as the fitness. The fitness of each individual is calculated, and the best individual is recorded. Selection, crossover, mutation, and constraint repair operations are performed on the population to update the individuals in the population, and the process is iterated to find the global optimal solution c with the minimum number of trap sets. i ={c0,c1,…,c L-1};

[0034] ③ Use the sequence c obtained in step ② i Construct the matrix E(H) shown in equation (3);

[0035] Step 3: Replace the 0 elements in the exponent matrix with an identity matrix of size q×q. The non-zero elements are the shift coefficients of the cyclic permutation matrix, that is, the identity matrix of size q×q is cyclically shifted to the right by the corresponding number of bits. After expansion, we get a parity check matrix H of size 3q×Lq.

[0036] In summary, there is no four-ring between the first and second rows of the original exponent matrix A(H), while the sequence c obtained by the genetic algorithm... i This ensures that there are no four- or six-ring loops between the third row and the first and second rows, and minimizes the number of trap sets in the resulting parity check matrix. Therefore, a short-code-length QC-LDPC(3,L) code with a ring length of 8 can be constructed using Stanley sequences combined with a genetic algorithm, and the code length and code rate can be flexibly selected. This type of code is named SS-QC-LDPC code.

[0037] 2. In conjunction with the appendix Figure 2 Note that the Tanner graph corresponding to the parity check matrix of the short code length QC-LDPC code constructed in this invention does not contain four-ring and six-ring structures.

[0038] (I) Proof that the first and second rows of the exponential matrix do not contain a four-ring structure.

[0039] Appendix Figure 2The structure of the Tanner graph with four rings is given. According to the ring length theorem and the properties of Stanley sequence, it can be proved that there are no four rings between the first and second rows.

[0040] Proof: The existence of the 4-ring is shown in the attached diagram. Figure 2 , where i0 < i1, j0 < j1.

[0041] If there are four rings in the Tanner graph corresponding to the verification matrix, according to the ring length theorem formula, equation (4) must be satisfied.

[0042]

[0043] Since in the first two rows of the exponential matrix E(H) we have e(i,j)=(i+1)b j (modp), then equation (4) can be expressed as:

[0044]

[0045] Since in the Stanley sequence, b0 ≥ 0, and b0 < b1 < b2 < ... < b n-1 Equation (6) is obviously invalid and contradicts the condition of the four-ring structure. Therefore, the first two rows of the exponential matrix E(H) do not contain the four-ring structure.

[0046] (II) Proof that the exponential matrix has no four- or six-rings

[0047] Appendix Figure 2 The structure diagram with six rings in the Tanner graph is given, and the sequence c obtained according to the designed genetic algorithm is... i This avoids creating a four- or six-ring pattern with the elements in the first and second rows, as shown in the figure. Therefore, all elements in the exponent matrix E(H) do not satisfy equation (2), and the four- or six-ring pattern does not exist in the Tanner diagram of the parity check matrix of the designed QC-LDPC code.

[0048] 3. Combined with the appendix Figure 3 To verify that the constructed codewords have superior error correction performance, MATLAB simulations were performed. Consider constructing a 3×5 exponential matrix, L=5, with an expansion factor q of 83. The designed search algorithm is used to obtain the sequence c. i ={5,67,42,33,68}. The constructed exponent matrix is ​​shown in equation (7). By expanding the exponent matrix, the SS-QC-LDPC(415,166) code constructed in this paper can be obtained.

[0049]

[0050] The error correction performance of the constructed SS-QC-LDPC code was verified by simulation using MATLAB software. The simulation environment was an additive white Gaussian noise (AWGN) channel, the modulation method was binary phase shift keying (BPSK), and the decoding was performed using the belief propagation (BP) algorithm. The maximum number of decoding iterations is 50. The SS-QC-LDPC(415,166) code constructed by this method is compared with the regular HS-QC-LDPC(415,166) code with a ring length of 8 constructed based on Hoey sequence (HS) in reference [1] "Yuan Jianguo, Song Wanchuang. A method for constructing 8-ring QC-LDPC codes based on Hoey sequence [J]. Telecommunications Technology, 2025, 65(05):793-799." and the modified Tanner (Modified) code constructed based on Hoey sequence (HS) in reference [2] "Kim W, Cho H, Song HY, Song MK. Some variations of Tanner's construction for short length QC-LDPC codes [J]. Electronics Letters. 2024, 60(3):e13088." The performance of the regular MT-QC-LDPC(415,166) code constructed by tanner (MT) and the regular SES-QC-LDPC(413,177) code based on the improved SES algorithm are compared. [3] "A. Ovinnikov, A. Kharin, and K. Zavertkin et al. Small Elementary Trapping Sets Elimination of Girth-8 QC-LDPC Codes[C]. Digital Signal Processing and its Applications, Moscow, Russian Federation, 2024:1-5."

[0051] The following is a comparison of the simulation curves between the bit error rate (BER) and signal-to-noise ratio (SNR) of different QC-LDPC codes under the same parameters. Figure 3 As shown. (From the appendix) Figure 3 It can be seen that when BER is 10 -6At that time, the net coding gain of the SS-QC-LDPC(415,166) code constructed by the present invention was improved by approximately 0.27, 0.42 and 0.51 dB compared with the HS-QC-LDPC(415,166) code, MT-QC-LDPC(415,166) code and SES-QC-LDPC(413,177) code, respectively.

[0052] Based on the simulation verifications above, this invention proposes a method for constructing short-code-length QC-LDPC codes by utilizing the special properties of Stanley sequences and combining them with a genetic algorithm. The Tanner graph of the parity check matrix corresponding to this codeword does not contain four- or six-ring loops and has a relatively small number of trap sets. Simulation results show that, under the same conditions, the SS-QC-LDPC code constructed in this invention exhibits improved net coding gain compared to HS-QC-LDPC codes, MT-QC-LDPC codes, and SES-QC-LDPC codes with the same code rate and code length, demonstrating better error correction performance and no obvious error flattening phenomenon.

[0053] Finally, it should be noted that the above preferred embodiments are only used to illustrate the technical solutions of the present invention and are not intended 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 to it in form and detail without departing from the scope defined by the claims of the present invention.

Claims

1. A method for constructing short-code-length QC-LDPC (Quasi-Cyclic Low-DensityParity-Check, QC-LDPC) codes based on Stanley sequences and genetic algorithms; this method first uses Stanley sequences to construct the first two rows of an exponent matrix that satisfies the condition of no 4-rings, then uses a genetic algorithm to search for the third row of the exponent matrix that satisfies the condition of a ring length of at least 8 and a small number of trap sets, constructing the corresponding exponent matrix. The genetic algorithm can guarantee that the number of trap sets is minimized. Finally, the exponent matrix is ​​expanded to obtain short-code-length QC-LDPC codes, so that the Tanner graph of the parity-check matrix does not contain 4-rings or 6-rings and the number of trap sets is small, the codeword requires less storage space, and it is easy to implement in hardware; the specific construction steps are as follows: Step 1: Construct the first two rows of the QC-LDPC(3,L) code exponent matrix A(H) that satisfies the 4-ring-free condition using the Stanley sequence, as shown in equation (1); the elements of the first and second rows are obtained by using the Stanley sequence b. i ={b0,b1,…,b L-1 We obtain that b0 < b1 < ... < b L-1 This can be expressed as A(i,j)=(i+1)b j , where 0≤i≤1, 0≤j≤L-1; the elements in the third row of the exponent matrix are represented by -1, i.e., A(2,j)=-1,0≤j≤L-1; Step 2: Combining Lemma 1, the Ring Length Theorem, and the properties of trap sets, design a genetic algorithm to search and obtain a set c. i ={c0,c1,…,c L-1 }, where all elements in set c are non-negative integers; set c i The elements in A(H) are replaced to obtain E(H), so that there are no four- or six-ring loops in the check matrix and only a small number of trap sets. E(H) is shown in equation (3). Lemma 1: In a QC-LDPC(m,n) code, the necessary and sufficient condition for the existence of a 2c ring is given by equation (2): (2) Where: j0=j n i l ≠i l+1 j l ≠j l+1 , 0≤i l <m, 0≤j l <n, where 0≤l≤c-1; Since there are no 4-rings between the second and third rows of A(H), this genetic algorithm only needs to ensure that there are no 4- or 6-rings between the third row and other rows while reducing the trap set; The specific steps of the genetic algorithm are as follows: ① Determine the size of the expansion factor q based on the required code length; ② The search interval is [0, q]. Within this interval, sequences that do not satisfy the existence theorem of the ring length in equation (2) are searched as individuals in the population. The number of trap sets is introduced as the fitness. The fitness of each individual is calculated, and the best individual is recorded. Selection, crossover, mutation, and constraint repair operations are performed on the population to update the individuals in the population. The process is iterated to find the global optimal solution c with the fewest trap sets. i ={c0,c1,…,c L-1 }; ③ Use the sequence c obtained in step ② i Construct the exponential matrix E(H) shown in equation (3); Step 3: Replace the 0 elements in the exponent matrix with an identity matrix of size q×q. The non-zero elements are the shift coefficients of the cyclic permutation matrix, that is, the identity matrix of size q×q is cyclically shifted to the right by the corresponding number of bits. After expansion, we get a parity check matrix H of size 3q×Lq. In summary, there is no four-ring between the first and second rows of the original exponent matrix A(H), while the sequence c obtained by the genetic algorithm... i This ensures that there are no four- or six-ring loops between the third row and the first and second rows, and minimizes the number of trap sets in the resulting parity check matrix. Therefore, by combining Stanley sequences with genetic algorithms, a short code length QC-LDPC(3,L) code with a ring length of 8 can be constructed, and the code length and code rate can be flexibly selected.

Images