Design of Nonbinary LDPC Cycle Codes with Large Girth from Circulants and Finite Fields

Xu, Hengzhou; Li, Huaan; Gao, Jixun; Zhang, Guixiang; Zhu, Hai; Zhang, Xiao-Dong

doi:https://doi.org/10.1155/2021/4099918

Wireless Communications and Mobile Computing

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Architectures, Challenges and Opportunities within 6G Emerging Technologies

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Research Article | Open Access

Volume 2021 | Article ID 4099918 | https://doi.org/10.1155/2021/4099918

Design of Nonbinary LDPC Cycle Codes with Large Girth from Circulants and Finite Fields

Hengzhou Xu,^1,2Huaan Li,¹Jixun Gao,³Guixiang Zhang,¹Hai Zhu,¹and Xiao-Dong Zhang²

Academic Editor: Miaowen Wen

Received21 May 2021

Accepted08 Sept 2021

Published23 Sept 2021

Abstract

In this paper, we study a class of nonbinary LDPC (NBLDPC) codes whose parity-check matrices have column weight 2, called NBLDPC cycle codes. We propose a design framework of -regular binary quasi-cyclic (QC) LDPC codes and then construct NBLDPC cycle codes of large girth based on circulants and finite fields by randomly choosing the nonzero field elements in their parity-check matrices. For enlarging the girth values, our approach is twofold. First, we give an exhaustive search of circulants with column/row weight and design a masking matrix with good cycle distribution based on the edge-node relation in undirected graphs. Second, according to the designed masking matrix, we construct the exponent matrix based on finite fields. The iterative decoding performances of the constructed codes on the additive white Gaussian noise (AWGN) channel are finally provided.

1. Introduction

Nonbinary low-density parity-check (NBLDPC) codes based on modulo arithmetics were first discovered by Gallager in 1960s [1] and redefined over finite fields GF by Davey and MacKay in 1998 [2]. Similar to binary LDPC codes, NBLDPC codes also have the ability of approaching capacity when decoded with the iterative algorithms. Moreover, NBLDPC codes have much better performance than binary LDPC codes for the short and moderate code lengths. As much more low-complexity decoding algorithms were proposed [3–8], NBLDPC codes provide a promising coding scheme for 6G communications [9].

As shown in [10], NBLDPC codes over larger finite fields will have much better performance for a constant code length. However, when the finite field size is sufficiently large, the performance improvement is little. Moreover, when the finite field size is equal or greater than 64, the column weights of the parity-check matrices of good NBLDPC codes tend to 2. Since NBLDPC cycle codes perform well over various channels [11–13], it is worth studying NBLDPC codes over large finite fields whose parity-check matrices have column weight 2, referred to as NBLDPC cycle codes. As an important cycle codes, -regular NBLDPC codes also perform well under iterative decoding; lots of methods for constructing such codes were proposed [14–17]. Among these works on the construction of NBLDPC codes, the codes can be mainly classified into two categories: the first one is constructed by means of computer search under the algorithms satisfying certain rules, and the other one is constructed based on combinatorial designs, graph theory, matrix theory, and finite fields [18]. Simulation results show that they all have good performance. For a given code rate and length, it is of great interest to study which one of them has the best error performance.

Cycle structure plays an important role in binary/nonbinary LDPC codes. Research results show that NBLDPC codes with large girth have good iterative performance [19]. In general, NBLDPC codes with large girth have large Hamming minimum distance, and it can be ensured that NBLDPC codes have good performance in the waterfall and error-floor region. Hence, it is interesting to construct LDPC cycle codes with large girth.

In this paper, we focus on the construction of -regular quasi-cyclic LDPC (QC-LDPC) codes with large girth. We first proposed the construction framework of -regular QC-LDPC codes based on the edge-node relation in undirected graphs and transfer the construction of -regular QC-LDPC codes into two main parts, i.e., circulants and exponent matrices. In the first part, we find circulants with good cycle distribution by performing an exhaustive search. In order to prune the search space of circulants, isomorphism theory of circulants is proposed. For the second part, we directly employ finite fields to construct exponent matrices of QC-LDPC codes. Here, the employed finite fields are divided two types, i.e., prime fields and finite fields of characteristic 2. Finally, numerical results to show the good performance of our proposed codes are provided.

The rest of this paper is organized as follows. Section 2 introduces the definitions of LDPC codes and their associated Tanner graphs. Section 3 presents the design framework of -regular QC-LDPC codes. Design of NBLDPC cycle codes with large girth is proposed in Section 4, and numerical results are also provided in this section. Finally, Section 5 concludes this paper.

2. Preliminaries

2.1. LDPC Codes

A binary -regular LDPC code is generated by the null space of an sparse parity-check matrix over GF (2), and the matrix has the following properties: (1) each column has 1’s; (2) each row has 1’s; (3) and . If the sparse matrix is over GF () for being a prime power, then LDPC codes generated by such are called nonbinary codes or -ary codes. Binary LDPC codes are referred to as quasi-cyclic (QC) [20], if their parity-check matrices have the following form where for , is a circulant permutation matrix (CPM) formed by cyclically shifting each row of a identity matrix to the right (or left) by positions, and is a zero matrix of size . Obviously, is an identity matrix of size . For example, if , then

We can easily see that the positions of 1’s of in (1) are uniquely determined by the following matrix , called exponent matrix (or permutation shift matrix),

It is not hard to see that the correspondence between and is one-to-one. It is noticeable that the parameter is called expansion factor (or lifting degree) [21]. By replacing 1’s in a CPM of in (1) with the same nonzero field element in finite field GF (), the resulting code is nonbinary QC-LDPC codes [22].

2.2. Tanner Graph

Apart from the matrix representation, an LDPC code can be also described in a simple and intuitive way, i.e., a graphical model called Tanner graph [23]. In fact, the Tanner graph of an LDPC code with the parity-check matrix is a bipartite graph in which the two classes of nodes are variable nodes (representing the code-bit nodes) and check nodes (representing the constraint nodes), respectively. An edge in a Tanner graph connects the check node to the variable node if and only if the row - and column - element in is nonzero. A cycle in a Tanner graph is a sequence of the connected check nodes and variable nodes which start and end at the same node in the graph and contain no other nodes more than once. The cycle length is simply the number of the contained edges (or nodes), and the length of the shortest cycle is referred to as girth of the Tanner graph (or an LDPC code). As an example, Figure 1 shows the Tanner graph of (or ) and an associate cycle of length 6 (6-cycle for short).

(a)

(b)

(c)

It is well-known that the iterative decoding algorithm converges to the optimal solution provided that the Tanner graph of an LDPC code is free of cycles [24]. In other words, short cycles, especially, the cycles of length 4, affect the decoding performance when decoded with the iterative algorithms based on belief propagation. In fact, there exist many cycles in an LDPC code with finite length. Hence, in order to avoid short cycles or obtain LDPC codes with large girth, many construction methods and techniques are proposed [25–33].

3. Design Framework of -Regular Binary QC-LDPC Codes

3.1. Edge-Node Relation in Undirected Graphs

Let be an undirected graph, where is a set of nodes and is some subset of the pairs (called edges) . A cycle of has distinct nodes (or edges), and an edge in a cycle has two distinct nodes. If we treat the nodes and edges of as the check nodes and variable nodes, respectively, then a bipartite graph can be obtained. Consider a cycle of length (denoted by -cycle for short) in . This -cycle will be turned into a -cycle in the above bipartite graph . In other words, the girth of is double that of . Based on this process, we can construct bipartite graphs (or Tanner graphs) with large girth from an undirected graph. In order to make it clearly, we give an example.

Consider the following matrix

It is easy to plot the Tanner graph of , given in Figure 2. By treating the nodes and edges of the Tanner graph in Figure 2 as the check nodes and variable nodes, respectively, we can construct a new bipartite graph, given in Figure 3. We can see from Figures 2 and 3 that a 4-cycle in the Tanner graph of becomes a 8-cycle in the bipartite graph.

3.2. Construction Framework of -Regular Binary QC-LDPC Codes

In this subsection, we will present the framework for constructing -regular binary QC-LDPC codes by using the edge-node relation in an undirected graph in Section 3.1. In order to design -regular codes, the node degree of should be . Furthermore, to guarantee -regular codes are quasi-cyclic, the incidence matrix of should possess quasi-cyclic structure. In conclusion, the incidence matrix of is -regular and quasi-cyclic. Hence, in order to obtain -regular binary QC-LDPC codes with large girth, we need to design a -regular quasi-cyclic matrix with large girth. For convenience, this -regular quasi-cyclic matrix is called base matrix. Next, we will give the construction framework.

First, we design a -regular base matrix of size . By employing the edge-node relation in Section 3.1, we can transfer the Tanner graph of into a new bipartite graph, and the incidence matrix of such a bipartite graph is obtained. It is obvious that is a -regular quasi-cyclic matrix of size . Second, we construct an exponent matrix of size , and the corresponding expansion factor is . Third, we use to mask the exponent matrix , and a array of CPMs is constructed. The null space of gives a -regular binary QC-LDPC code of length .

4. Design of Nonbinary LDPC Cycle Codes with Large Girth

In order to construct -regular binary QC-LDPC codes with large girth, we only design a base matrix and a corresponding exponent matrix based on the construction framework in Section 3.2. By replacing the nonzero element in the parity-check matrices of binary QC-LDPC cycle codes with the nonzero field elements, nonbinary LDPC cycle codes can be obtained. In this paper, we do not optimize the nonzero field elements and adopt the optimized row elements in [34]. Next, we will provide the construction of the base matrices and exponent matrices.

4.1. Exhaustive Search of Circulants Based on Isomorphism Theory

In this paper, we employ the circulant as the base matrix. It can be seen from the construction framework in Section 3.2 that the size of the base matrix is not too large since the code lengths of NBLDPC codes are short or moderate. In the following, we will give the design of the circulants.

A circulant is a square matrix whose -th row is generated by cyclically shifting the first row to the right (or left) by positions. Hence, the first row of a circulant is referred to as the generator of the circulant. For a circulant of size , each row (or column) is a rightward (or downward) cyclic-shift of its above (or left) row (or column), and the first row (or column) is the rightward (or downward) cyclic-shift of the last row (or column). Therefore, the rows and columns of a circulant have the same weight. It is clear that the row (or column) weight is associated with the row weight of the generator.

Consider a circulant of size , and its generator is where for . Let be the number of the nonzero components of . Hence, is -regular. We select the nonzero components from and record their subscripts in a set , called location set in this paper. Then, the location set has elements. Without loss of generality, the location set is denoted by where for . It is obvious that the generator and the location set have a one-to-one correspondence. Based on the isomorphism theory of LDPC codes (or their parity-check matrices) in [16, 35, 36], we can directly give the isomorphism theory of the circulants as follows.

Theorem 1. Let and be two location sets of the circulants and of size , respectively. Then, is isomorphic to , denoted by , if is derived from with either of the following two methods. (1)For , the elements of are derived from these of by adding a constant to the elements of modulo , i.e., for (2)Suppose that and are coprime. The elements of are derived from these of using for

Note that in the calculation process, if the element in and equals 0, it actually equals . Moreover, in the case (2) of Theorem 1, the number of can be determined by a well-known function, called Euler’s phi function, i.e.,

If , we say is isomorphic to , denoted by .

In general, the size of the employed circulants in this paper is not large. Hence, we can make an exhaustive search of the circulants by using the computer. The search space of the location sets of the circulants with row/column weight is

Based on the case (1) in Theorem 1, we can see that all the location sets of the circulants have the following isomorphic form: , where for . Hence, the search space of such location sets is

That is, an exhaustive search of such location sets (or circulants) is feasible. Here, we do not provide the specific exhaustive search algorithm. Combined with the cycle-counting algorithms [37–39], the optimal circulants with row/column weight can be found. In this paper, the optimal ones are such circulants whose Tanner graphs have fewer short cycles and larger girths. In order to facilitate the readers, some optimal circulants are presented in Table 1.

4.2. Review of Finite Fields Based QC-LDPC Codes and Their Exponent Matrices

In this subsection, we will review the finite field based method for constructing QC-LDPC codes, and provide two classes of exponent matrices of these QC-LDPC codes [18].

4.2.1. A General Construction of QC-LDPC Codes Based on Finite Fields of Characteristic 2

Let GF be a finite field with with , and let be a primitive element of GF . For each nonzero element with , we define the location vector as a -tuple over GF (2), whose components correspond to the nonzero elements of GF , where the -th component is set to 1 and all the other components are 0. Hence, based on the nonzero element of GF , we can uniquely form a square matrix whose -th row is obtained by cyclically shifting every component of the -ary location vector places to the right (or left) for . The resulting square matrix is a CPM, and it is also referred to as the -fold matrix dispersion (or expansion) of the nonzero field element over GF (2) [18].

Consider a matrix over GF , whose rows satisfy the following two constraints: (1) for and , and have at most one position where both of them have the same element of GF (i.e., they differ in at least positions); (2) for and and differ in at least positions. These constraints are called -multiplied row constraints in [18]. By replacing each of its entry with the binary matrix , we obtain a array of CPMs. Then, the null space of gives a binary -regular QC-LDPC code. Moreover, the -multiplied row constraints ensure the Tanner graph of this code free of cycle of length 4. Hence, the constructed QC-LDPC codes have girth at least 6.

According to the definition in Section 2, we can see that the above matrix is the exponent matrix, and the associate expansion factor is . A framework for constructing such matrix based on two arbitrary subsets of a finite field was proposed in [40]. Let be a nonzero element in GF and be a primitive element. For , let and be two arbitrary subsets of elements in GF with and in the set and . The matrix can be formed by

Under this framework, some well-known constructions of QC-LDPC codes based on finite fields and combinatorial designs are special cases [14, 41, 42]. For example, when , is a Latin square over GF [43].

4.2.2. Construction of QC-LDPC Codes Based on Prime Fields

Let be a prime. Consider a matrix where for and . We select a submatrix from and replace its elements with the CPMs for and . The following array of CPMs over GF (2) is obtained.

Actually, the null space of gives a -regular QC-LDPC code of length with girth at least 6. Notice that the exponent matrix of this code is and the expansion factor is . How to select good rows and columns of CPMs from in (12) can be found in [44, 45].

4.3. Nonbinary LDPC Cycle Codes and Numerical Results

Combined with Sections 3.2, 4.1, and 4.2, we can easily construct a binary QC-LDPC cycle code. Based on the replacement of the nonzero elements in finite fields, nonbinary LDPC cycle codes can be constructed. In order to show the advantages of our proposed construction methods, we next provide some examples.

Example 1. Consider the all-one matrix (or circulant) of size . It is clear that there is only one such circulant for and . The following matrix can be obtained based on the construction framework in Section 3.2. Based on the prime field GF (31), we can construct an array of CPMs in the form of (13) by choosing the 8 rows and 16 columns from the exponent matrix in equation (12). The indices of the chosen 8 rows and 16 columns from are {3, 4, 5, 10, 15, 17, 24, 28} and {1, 4, 5, 6, 7, 8, 9, 10, 11, 15, 16, 19, 26, 28, 29, 30}, respectively. The row and column selection method is based on the proposed method in [45]. By employing to mask , we can obtain an array of CPMs. By replacing the 1’s of each row in with nonzero elements of GF (64) in the corresponding rows of Table 2, a -regular matrix over GF (64) is obtained. Note that the numbers in Table 2 are the power numbers of , where is a primitive element of GF (64) created by using the primitive polynomial . The null space of gives a -regular (496,248) LDPC cycle code over GF (64). The girth of this code is 16, and the number of 16-cycles is 775. For comparison, we reconstruct the irregular (504,252) QC-LDPC code over GF (64) based on finite field GF (64) and the masking matrix in [15]. The average column and row weights in the parity-check matrix of (504,252) QC-LDPC code over GF (64) are 2.5 and 5, respectively. Moreover, we also design a comparable (2,4)-regular (496,248) LDPC code over GF (64) based on the progressive edge-growth (PEG) algorithm in [46], called PEG-LDPC code. Note that the nonzero elements of the parity-check matrix of the (2,4)-regular (496,248) PEG-LDPC code over GF (64) are randomly chosen. When decoded with the fast-Fourier-transform (FFT) based Q-ary sum-product algorithm (QSPA) with 50 iterations, the bit/word error rates (BERs/WERs) of these three nonbinary LDPC codes are shown in Figure 4. In the simulations, the transmissions on the BPSK modulated AWGN channel are assumed. We can see that at the BER of , the proposed nonbinary (496,248) LDPC cycle code outperforms the (504,252) QC-LDPC code over GF (64) and the (2,4)-regular (496,248) PEG-LDPC code over GF (64) about 0.45 dB and 0.1 dB, respectively.

In the above example, we can see that the constructed nonbinary LDPC cycle code is not quasi-cyclic because of the randomly chosen nonzero field elements in its parity-check matrix. Hence, we next show the performances of the proposed nonbinary QC-LDPC cycle codes.

Example 2. Consider the prime field GF (19), we can easily construct an matrix over GF (19) based on the exponent matrix in equation (12) and the construction framework in Section 3.2. The indices of the chosen 8 rows and 16 columns from are {1, 2, 4, 5, 7, 8, 14, 15} and {1, 2, 3, 4, 5, 6, 7, 9, 10, 13, 14, 15, 16, 17, 18, 19}, respectively. The row and column selection method is also based on the proposed method in [45]. By dispersing each entry of into the corresponding CPMs of size , an array of CPMs is obtained. By employing in equation (14) to mask the array , we can construct an array of CPMs. By replacing the 1’s of each CPM in with the same nonzero element of GF (256) in the corresponding row of Table 3, a (2,4)-regular matrix over GF (256) is obtained. It is noticeable that the numbers in Table 3 are the power numbers of , where is a primitive element of GF (256) created by using the primitive polynomial . The null space of gives a (2,4)-regular (304,152) LDPC cycle code over GF (256). The girth of this code is 16, and the number of 16-cycles is 969. Since the nonzero field elements of a CPM in are the same, and the resulting (304,152) LDPC cycle code over GF (256) is quasi-cyclic.
For comparison, we construct the (3,6)-regular (2432,1216) LDPC code over GF (2) based on the progressive edge-growth (PEG) algorithm in [46]. The error performances of these two LDPC codes are shown in Figure 5. In the simulations, the employed decoding algorithms of the constructed (304,152) QC-LDPC cycle code over GF (256) and the (2432,1216) LDPC code over GF (2) are the QSPA (50 iterations) and the sum-product algorithm (SPA) with 50 iterations, respectively. The transmissions on the BPSK modulated AWGN channel are assumed. We can see that the constructed (304,152) QC-LDPC cycle code over GF (256) can outperform the (2432,1216) LDPC code over GF (2) about 0.75 dB at the BER of .

5. Conclusion

This paper proposed a design framework of binary QC-LDPC cycle codes and constructed nonbinary LDPC (NBLDPC) cycle codes based on circulants and finite fields. The presented construction method consists of three parts. First, the masking matrices are designed based on circulants and the point-line relation in graph theory. Second, the exponent matrices of binary QC-LDPC cycle codes are constructed from finite fields and the designed masking matrices. Third, by replacing 1’s in the parity-check matrices of binary QC-LDPC cycle codes with the nonzero field elements, NBLDPC cycle codes are obtained. Numerical results show that the constructed NBLDPC cycle codes have good iterative decoding performance.

Data Availability

The data used to support the findings of this study is available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants 61801527 and 11971311, the TianYuan Special Funds of the National Natural Science Foundation of China under Grants 12026230 and 12026231, and the Development Project of Henan Provincial Department of Science and Technology under Grants 212102310544 and 212102310551.

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Copyright © 2021 Hengzhou Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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