Short code length 8-ring QC-LDPC code construction method based on FLSS algorithm

Abstract

The invention relates to a short code length eight-ring QC-LDPC (Quasi-Cyclic Low Density Parity-Check) code construction method based on an FLSS algorithm. According to the method, firstly, a Fibonacci-Lucas sequence is utilized to form an index matrix meeting a four-ring-free condition, then an FLSS algorithm is utilized to search out elements enabling the index matrix to meet the condition that the girth is at least 8, an incremental sequence is obtained, the index matrix with the girth being at least 8 is obtained, the FLSS algorithm can guarantee that selection of expansion factors has flexibility, and the expansion efficiency is improved. 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 required storage space is small, and hardware implementation is easy. A simulation result shows that compared with a PM-QC-LDPC code, a GR-QC-LDPC code, an IS-QC-LDPC code and a DDS-QC-LDPC code which have the same code rate and code length, the FLSS-QC-LDPC code constructed by the method has the advantages that the net coding gain is improved to a certain degree, and the constructed FLSS-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 a short code length 8-ring Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) code based on a Fibonacci-Lucas sequence and search (FLSS) algorithm. 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 (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 short code lengths have the advantages of low latency, simple implementation, and suitability for real-time applications. Therefore, the design of short code length QC-LDPC codes is of great significance in the physical layer design of high-real-time and high-reliability communications.

[0004] The construction methods of QC-LDPC codes 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 use algebraic construction methods. The present invention proposes a new construction method based on the algebraic construction method, first using the Fibonacci-Lucas sequence to form an exponential matrix that satisfies the condition of no 4 rings, and then using the FLSS algorithm to make its exponential matrix satisfy the girth of at least 8, constructing the corresponding exponential matrix, and obtaining its parity check matrix, so that there are no four rings and six rings in the Tanner graph of the check matrix, and finally designing a class of short code length QC-LDPC codes with row weight L, column weight J, and 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 a method for constructing a QC-LDPC code with a short code length of 8 rings based on the FLSS algorithm. The method first uses a Fibonacci-Lucas sequence to construct an exponential matrix that satisfies the condition of no 4 rings, and then uses the FLSS algorithm to make its exponential matrix satisfy the girth of at least 8, constructs the corresponding exponential matrix, and finally obtains 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, the QC-LDPC (3, L) code index matrix A (H) that satisfies the 4-cycle-free condition is constructed through the Fibonacci-Lucas sequence. The first row of the index matrix is ​​all 0, that is, A (0, s) = 0, 0 ≤ s ≤ L-1. The elements of the second and third rows are obtained through the Fibonacci-Lucas sequence, expressed as A (i, s) = F (2i + s), where 1 ≤ i ≤ 2, 0 ≤ s ≤ L-1.

[0008] Then, combining the ring length theorem and the characteristics of four-six rings, a FLSS algorithm is designed to construct the sequence b i ={b0,b1,…,b L-1}, and b0<b1<…<b L-1 , and then sequence b i ={b0,b1,…,b L-1}Replace the first row of A(H) to get E(H). Because there is no 4-ring between the second and third rows of A(H), the FLSS algorithm only needs to ensure that there is no 4-ring or 6-ring between the first row searched and other rows.

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

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

[0011] ② Determine the search interval [1,z], where z<q; in order to reduce the difficulty of the search, consider first determining the sequence b in the interval [1,30] i The first two or three elements of , so that these elements and the second and third rows do not satisfy the four-six ring existence theorem; then search the remaining elements in the interval [31, z] in turn to form the sequence b i ={b0,b1,…,b L-1}, b0<b1<…<b L-1 .

[0012] ③Use the sequence b obtained in step ② i Construct the matrix E(H) and check whether E(H) has a four-six ring structure, that is, whether all elements in E(H) do not satisfy the four-six ring existence theorem; if they do not satisfy it, determine the sequence bi ={b0,b1,…,b L-1}, otherwise repeat step ②.

[0013] Finally, the 0 elements in the exponential matrix are replaced by the identity matrix of size q×q, and the non-zero elements are the shift coefficients of the circulant permutation matrix, that is, the identity matrix of size q×q is circulantly 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, through the above 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 are constructed. In the same simulation environment, a short code length 8-ring QC-LDPC code construction method based on the FLSS algorithm proposed in this patent is simulated and compared with other similar coding schemes.

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

[0016] The present invention relates to a method for constructing a short code length 8-ring QC-LDPC code based on an FLSS algorithm. 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 ring length of the constructed check matrix is ​​at least 8, and it has excellent error correction performance and no obvious error flattening phenomenon. Simulation results show that the FLSS-QC-LDPC (684, 342) code constructed by this method is comparable to the IS-QC-LDPC (684, 342) code constructed based on integer sequence and special structure in literature [1] "TAO X, CHEN X, WANG B. On the Construction of QC-LDPC Codes Based on Integer Sequence With Low Error Floor [J]. IEEE Communications Letters, 2022, 26 (10): 2267-2271.", the regular GR-QC-LDPC (684, 342) code constructed based on Golomb Ruler in literature [2] "KIM I, SONG H YA construction forgirth-8QC-LDPC codes using Golomb rulers [J]. Electronics Letters, 2022, 58 (15): 582-584.", and the regular GR-QC-LDPC (684, 342) code constructed based on Golomb Ruler in literature [3] "KIM I, KOJIMA T, SONG H Y. Some Short-Length Girth-8 QC-LDPC Codes From Primes of the Form t 2 +1[J].IEEE Communications Letters,2022,26(6):1211-1215." Compared with the regular PM-QC-LDPC (684,342) code constructed based on prime numbers and multiplication tables, 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-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 1A 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 short code length 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 a short code length 8-ring QC-LDPC code based on the FLSS algorithm is as follows:

[0023] Step 1: Construct a QC-LDPC (3, L) code index matrix A(H) that satisfies the 4-cycle-free condition through the Fibonacci-Lucas sequence, as shown in formula (1); the first row of the index matrix is ​​all 0, that is, A(0,s)=0,0≤s≤L-1; the second and third rows of elements are obtained through the Fibonacci-Lucas sequence, expressed as A(i,s)=F(2i+s), where 1≤i≤2,0≤s≤L-1;

[0024]

[0025] Step 2: Combining the ring length theorem and the characteristics of four-six rings, design a specific FLSS algorithm to construct sequence b i ={b0,b1,…,b L-1}, and b0<b1<…<b L-1 , and then sequence b i ={b0,b1,…,b L-1}Replace the first row of A(H) to obtain E(H), so that E(H) does not satisfy Lemma 1, and E(H) is as shown in formula (3);

[0026] Lemma 1: In QC-LDPC (m, n) codes, the necessary and sufficient condition for the existence of a 2c ring is (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] The specific steps of the FLSS algorithm are as follows:

[0031] ① Determine the size of the expansion factor q according to the required code length;

[0032] ② Determine the search interval [1,z], where z<q; in order to reduce the difficulty of the search, consider first determining the sequence b in the interval [1,30] i The first two or three elements of the sequence b are selected so that the ring length between these elements and the second and third rows does not satisfy the ring length existence theorem of formula (2); then the remaining elements are searched in the interval [31, z] in turn to form the sequence b i ={b0,b1,…,b L-1}, b0<b1<…<b L-1 ;

[0033] ③Use the sequence b obtained in step ② i Construct the matrix E(H) shown in formula (3) and check whether E(H) has a four-six ring structure, that is, whether all elements in E(H) do not satisfy formula (2); if they do not satisfy formula (2), determine the sequence b i ={b0,b1,…,b L-1}, otherwise repeat step ②;

[0034] Step 3: Replace the 0 elements in the exponential matrix with a unit matrix of size q×q. The non-zero elements are the shift coefficients of the circulant permutation matrix, that is, the unit matrix of size q×q is circulantly shifted to the right by the corresponding number of bits. After expansion, the parity check matrix H of size 3q×Lq is obtained.

[0035] In summary, there is no quadruple loop between the second and third rows in the original exponential matrix A(H), and the sequence b obtained by the FLSS algorithm i Obviously, it is guaranteed that there is no four-six loop between the first row and the second and third rows. Therefore, when z < q, a class of short code length QC-LDPC (3, L) codes with a girth of 8 can be constructed by selecting a reasonable expansion factor q value, and the code length and code rate can be flexibly selected. This type of code is named FLSS-QC-LDPC code.

[0036] 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 parity check matrix of the short-code long QC-LDPC code constructed by the present invention.

[0037] (I) Proof that the second and third rows of the exponential matrix do not have a quadruple ring

[0038] Attached Figure 2The structure of the existence of four rings in the Tanner graph is given in . According to the ring length theorem and the characteristics of the Fibonacci-Lucas sequence, it can be proved that there is no four ring between the second and third rows.

[0039] Proof: The existence of 4 rings is as follows Figure 2 , where i1<i2,j1<j2.

[0040] a(i1,j1)-a(i1,j2)+a(i2,j2)-a(i2,j1)=F(2i1+j1)-F(2i1+j2)+F(2i2+j2)-F(2i2+j1)>0 (4)

[0041] a(i1,j1)-a(i1,j2)+a(i2,j2)-a(i2,j1)<a(i2,j2)-a(i2,j1)<a(i2,j2)<q (5)

[0042] From equations (4) and (5), we can obtain that e(i1,j1)-e(i1,j2)+e(i2,j2)-e(i2,j1)≠0modq, which means that the condition for the existence of a quadruple ring is not met. There is no quadruple ring in the second and third rows of the exponential matrix E(H).

[0043] (II) Proof that exponential matrices do not have four or six rings

[0044] Attached Figure 2 The structure diagram of the six rings in the Tanner graph is given, and the sequence b obtained by the designed FLSS algorithm is i This makes E(H) of equation (3) not satisfy Lemma 1. Therefore, when the shift coefficient q>z, all elements in the exponential matrix E(H) do not satisfy equation (2), and there is no four-six cycle in the Tanner graph of the parity check matrix of the designed QC-LDPC code.

[0045] 3. Combine with attachment Figure 3 In order to verify that the constructed codeword has better error correction performance, MATLAB is used for simulation. Consider constructing a QC-LDPC code with row weight L = 6 and column weight J = 3, the expansion factor q is 114, z is 70, and the search interval is [1,70], and the sequence b is obtained. i ={17,19,52,62,64,65}. The constructed exponential matrix is ​​shown in formula (6):

[0046]

[0047] The error correction performance of the constructed FLSS-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 FLSS-QC-LDPC (684, 342) code constructed by this method is compared with the IS-QC-LDPC (684, 342) code constructed based on integer sequence and special structure in literature [1] "TAO X, CHEN X, WANG B. On the Construction of QC-LDPC Codes Based on Integer Sequence With Low Error Floor [J]. IEEE Communications Letters, 2022, 26 (10): 2267-2271.", the regular GR-QC-LDPC (684, 342) code constructed based on Golomb Ruler in literature [2] "KIM I, SONG H YA construction for girth-8QC-LDPC codes using Golomb rulers [J]. Electronics Letters, 2022, 58 (15): 582-584.", and the regular GR-QC-LDPC (684, 342) code constructed based on Golomb Ruler in literature [3] "KIM I, KOJIMA T,SONG H Y.Some Short-LengthGirth-8 QC-LDPC Codes From Primes of the Form t 2 +1[J].IEEE Communications Letters, 2022, 26(6):1211-1215." Performance comparison of regular PM-QC-LDPC (684,342) codes constructed based on prime numbers and multiplication tables.

[0048] 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 3 It can be seen that when the BER is 10 -5The net coding gain (NCG) of the FLSS-QC-LDPC (684,342) code constructed by the present invention is improved by approximately 0.12, 0.39, and 0.62 dB respectively compared with the IS-QC-LDPC (684,342) code, GR-QC-LDPC (684,342) code, and PM-QC-LDPC (684,342) code.

[0049] Based on the above simulation verification, it can be known that the present invention uses the special properties of the Fibonacci-Lucas sequence and combines the FLSS algorithm to propose a short code length QC-LDPC code construction method. 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 FLSS-QC-LDPC code constructed by the present invention has a certain improvement in net coding gain NCG compared with the IS-QC-LDPC code, GR-QC-LDPC code and PM-QC-LDPC code with the same code rate and code length, and has good error correction performance and no obvious error flattening phenomenon.

[0050] 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 a short code length 8-ring QC-LDPC (Quasi-Cyclic Low-Density Parity-Check, QC-LDPC) code based on a Fibonacci-Lucas sequence and search (FLSS) algorithm; the method first uses the Fibonacci-Lucas sequence to form an exponential matrix that satisfies the condition of no 4-rings, and then uses the FLSS algorithm to search for elements whose exponential matrix satisfies a girth of at least 8 to obtain an increasing sequence, and constructs a corresponding exponential matrix. The FLSS algorithm can ensure that the selection of an expansion factor is flexible, and finally the exponential matrix is ​​expanded to obtain a short code length QC-LDPC code, so that there are no four rings and six rings in the Tanner graph of the check matrix, and the required storage space is small, which is easy to implement in hardware; the specific construction steps are as follows: Step 1: Construct a QC-LDPC (3, L) code index matrix A(H) that satisfies the 4-cycle-free condition through the Fibonacci-Lucas sequence, as shown in formula (1); the first row of the index matrix is ​​all 0, that is, A(0,s)=0,0≤s≤L-1; the second and third rows of elements are obtained through the Fibonacci-Lucas sequence, expressed as A(i,s)=F(2i+s), where 1≤i≤2,0≤s≤L-1; Step 2: Combining the ring length theorem and the characteristics of four-six rings, design a specific FLSS algorithm to construct sequence b i ={b0,b1,…,b L-1 }, and b0<b1<…<b L-1 , and then sequence b i ={b0,b1,…,b L-1 }Replace the first row of A(H) to obtain E(H), so that E(H) does not satisfy Lemma 1, and E(H) is as shown in formula (3); Lemma 1: In QC-LDPC (m, n) codes, the necessary and sufficient condition for the existence of a 2c ring is (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; Because there is no 4-ring between the second and third rows of A(H), the FLSS algorithm only needs to ensure that there is no 4-ring or 6-ring between the first row searched and other rows; The specific steps of the FLSS algorithm are as follows: ① Determine the size of the expansion factor q according to the required code length; ② Determine the search interval [1,z], where z<q; in order to reduce the difficulty of the search, consider first determining the sequence b in the interval [1,30] i The first two or three elements of the sequence b are selected so that the ring length between these elements and the second and third rows does not satisfy the ring length existence theorem of formula (2); then the remaining elements are searched in the interval [31, z] in turn to form the sequence b i ={b0,b1,…,b L-1 }, b0<b1<…<b L-1 ; ③Use the sequence b obtained in step ② i Construct the matrix E(H) shown in formula (3) and check whether E(H) has a four-six ring structure, that is, whether all elements in E(H) do not satisfy formula (2); if they do not satisfy formula (2), determine the sequence b i ={b0,b1,…,b L-1 }, otherwise repeat step ②; Step 3: Replace the 0 elements in the exponential matrix with a unit matrix of size q×q. The non-zero elements are the shift coefficients of the circulant permutation matrix, that is, the unit matrix of size q×q is circulantly shifted to the right by the corresponding number of bits. After expansion, the parity check matrix H of size 3q×Lq is obtained. In summary, there is no quadruple loop between the second and third rows in the original exponential matrix A(H), and the sequence b obtained by the FLSS algorithm i Obviously, it is guaranteed that there is no four-six loop between the first row and the second and third rows. Therefore, when z<q, by selecting a reasonable expansion factor q value, a class of short code length QC-LDPC (3, L) codes with a girth of 8 can be constructed, and the code length and code rate can be flexibly selected.

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