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

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 ð 2, ρ Þ -regular binary quasi-cyclic (QC) LDPC codes and then construct NBLDPC cycle codes of large girth based on circulants and ﬁ nite ﬁ elds by randomly choosing the nonzero ﬁ eld 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 ρ 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 ﬁ nite ﬁ elds. The iterative decoding performances of the constructed codes on the additive white Gaussian noise (AWGN) channel are ﬁ nally provided.


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 ðqÞ 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][4][5][6][7][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 per-form well over various channels [11][12][13], it is worth studying NBLDPC codes over large finite fields whose paritycheck matrices have column weight 2, referred to as NBLDPC cycle codes. As an important cycle codes, ð2, ρÞregular NBLDPC codes also perform well under iterative decoding; lots of methods for constructing such codes were proposed [14][15][16][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 ð2, ρÞregular quasi-cyclic LDPC (QC-LDPC) codes with large girth. We first proposed the construction framework of ð2, ρÞ-regular QC-LDPC codes based on the edge-node relation in undirected graphs and transfer the construction of ð2, ρÞ-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 ð2, ρÞ-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.

Preliminaries
2.1. LDPC Codes. A binary ðγ, ρÞ-regular LDPC code is generated by the null space of an m × n sparse paritycheck matrix H over GF (2), and the matrix H has the following properties: (1) each column has γ 1's; (2) each row has ρ 1's; (3) γ ≪ m and ρ ≪ n. If the sparse matrix H is over GF (q) for q being a prime power, then LDPC codes generated by such H are called nonbinary codes or q-ary codes. Binary LDPC codes are referred to as quasicyclic (QC) [20], if their parity-check matrices H have the following form respectively. An edge in a Tanner graph connects the check node s to the variable node t if and only if the row -s and column -t element h s,t in H 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 H b (or H nb ) and an associate cycle of length 6 (6-cycle for short).
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][26][27][28][29][30][31][32][33].

Design Framework of ð2, ρÞ-Regular
Binary QC-LDPC Codes 3.1. Edge-Node Relation in Undirected Graphs. Let G = ðV, EÞ be an undirected graph, where V is a set of nodes and E is some subset of the pairs (called edges) ffa, bg: a, b ∈ V, a ≠ bg. A cycle of G = ðV, EÞ has distinct nodes (or edges), and an edge in a cycle has two distinct nodes. If we treat the nodes and edges of G = ðV, EÞ as the check nodes and variable nodes, respectively, then a bipartite graph G B can be obtained. Consider a cycle of length k (denoted by k -cycle for short) in G = ðV, EÞ. This k-cycle will be turned into a 2k-cycle in the above bipartite graph G B . In other words, the girth of G B is double that of G = ðV, EÞ. 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. : ð4Þ It is easy to plot the Tanner graph of B 4×4 , 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 B 4×4 becomes a 8-cycle in the bipartite graph.

Construction
Framework of ð2, ρÞ-Regular Binary QC-LDPC Codes. In this subsection, we will present the framework for constructing ð2, ρÞ-regular binary QC-LDPC codes by using the edge-node relation in an undirected graph in Section 3.1. In order to design ð2, ρÞ-regular codes, the node degree of G = ðV, EÞ should be ρ. Furthermore, to guarantee ð2, ρÞ-regular codes are quasi-cyclic, the incidence matrix of G = ðV, EÞ should possess quasi-cyclic structure. In conclusion, the incidence matrix of G = ðV, EÞ is ðρ, ρÞ-regular and quasi-cyclic. Hence, in order to obtain ð2, ρÞ-regular binary QC-LDPC codes with large girth, we need to design a ðρ, ρÞ-regular quasi-cyclic matrix with large girth. For convenience, this ðρ, ρÞ-regular quasi-cyclic matrix is called base matrix. Next, we will give the construction framework.
First, we design a ðρ, ρÞ-regular base matrix B of size L × L. By employing the edge-node relation in Section 3.1, we can transfer the Tanner graph of B into a new bipartite graph, and the incidence matrix B M of such a bipartite graph is obtained. It is obvious that B M is a ð2, ρÞ-regular quasicyclic matrix of size 2L × ρL. Second, we construct an exponent matrix P of size 2L × ρL, and the corresponding expansion factor is Q. Third, we use B M to mask the exponent matrix P, and a 2L × ρL array H M of Q × Q CPMs is constructed. The null space of H M gives a ð2, ρÞ-regular binary QC-LDPC code of length ρLQ.  (4); (c) Tanner graph.

Design of Nonbinary LDPC Cycle Codes with Large Girth
In order to construct ð2, ρÞ-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.

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 i-th row is generated by cyclically shifting the first row to the right (or left) by ði − 1Þ positions. Hence, the first row of a circulant is referred to as the generator of the circulant. For a circulant of size L × L, 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 C of size L × L, and its generator is Let ρ be the number of the nonzero components of G. Hence, C is ðρ, ρÞ-regular. We select the nonzero components from g 1 , g 2 , ⋯, g L and record their subscripts in a set S, called location set in this paper. Then, the location set S has ρ elements. Without loss of generality, the location set S is denoted by It is obvious that the generator G and the location set S 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.
(1) For r ∈ f0, 1, ⋯, L − 1g, the elements of S 2 are derived from these of S 1 by adding a constant r to the elements of S 1 modulo L, i.e., s 2,i = s 1,i + rðmod LÞ for 1 ≤ i ≤ ρ (2) Suppose that r and L are coprime. The elements of S 2 are derived from these of S 1 using s 2,i = r · s 1,i ðmod LÞ Note that in the calculation process, if the element in S 1 and S 2 equals 0, it actually equals L. Moreover, in the case (2) of Theorem 1, the number of r can be determined by a wellknown function, called Euler's phi function, i.e., If C 1 ≅ C 2 , we say S 1 is isomorphic to S 2 , 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 L × L circulants with row/column weight ρ is Based on the case (1) in Theorem 1, we can see that all the location sets of the L × L circulants have the following isomorphic form: S = fs 1 ð= 1Þ, s 2 , ⋯, s ρ g, where 2 ≤ s i < s j ≤ L for 2 ≤ i < j ≤ ρ. Hence, the search space of such location sets S 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 cyclecounting algorithms [37][38][39], the optimal L × L circulants with row/column weight ρ 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.

Review of Finite Fields Based QC-LDPC Codes and Their
Exponent Matrices. In this subsection, we will review the l 1 l 2 l 3 l 4 l 5 l 6 l 7 l 8 l 9 l 10 l 11 l 12

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finite field based method for constructing QC-LDPC codes, and provide two classes of exponent matrices of these QC-LDPC codes [18].

A General Construction of QC-LDPC Codes Based on
Finite Fields of Characteristic 2. Let GF ðqÞ be a finite field with q = 2 t with t ≥ 2, and let α be a primitive element of GF ðqÞ. For each nonzero element α i with 0 ≤ i ≤ q − 2, we define the location vector vðα i Þ as a ðq − 1Þ-tuple over GF (2), whose components correspond to the nonzero elements α 0 , α 1 , ⋯, α q−2 of GF ðqÞ, where the i-th component v i is set to 1 and all the other ðq − 2Þ components are 0. Hence, based on the nonzero element α i of GF ðqÞ, we can uniquely form a ðq − 1Þ × ðq − 1Þ square matrix Mðα i Þ whose j-th row is obtained by cyclically shifting every component of the ðq − 1Þ-ary location vector vðα i Þðj − 1Þ places to the right (or left) for 0 ≤ i ≤ q − 2, 1 ≤ j ≤ q − 1. The resulting square matrix Mðα i Þ is a CPM, and it is also referred to as the ðq − 1Þ-fold matrix dispersion (or expansion) of the nonzero field element α i over GF (2) [18].

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.