Research Letter  Open Access
Jun Fan, Yang Xiao, Kiseon Kim, "Design LDPC Codes without Cycles of Length 4 and 6", Journal of Electrical and Computer Engineering, vol. 2008, Article ID 354137, 5 pages, 2008. https://doi.org/10.1155/2008/354137
Design LDPC Codes without Cycles of Length 4 and 6
Abstract
We present an approach for constructing LDPC codes without cycles of length 4 and 6. Firstly, we design 3 submatrices with different shifting functions given by the proposed schemes, then combine them into the matrix specified by the proposed approach, and, finally, expand the matrix into a desired paritycheck matrix using identity matrices and cyclic shift matrices of the identity matrices. The simulation result in AWGN channel verifies that the BER of the proposed code is close to those of Mackay's random codes and Tanner's QC codes, and the good BER performance of the proposed can remain at high code rates.
1. Introduction
LDPC codes can be described by a bipartite graph called Tanner graph [1], and the girth of a Tanner graph is the length of the shortest cycle in the graph. Girths in the Tanner graphs of LDPC codes prevent the sumproduct algorithm from converging [2–10]. Further, cycles, especially short cycles, degrade the performance of LDPC decoders, because they affect the independence of the extrinsic information exchanged in the iterative decoding [2, 3]. Hence, LDPC codes with large girth are desired. Most methods for designing LDPC codes are based on random construction techniques, the lack of structure implied by this randomness presents serious disadvantages in terms of storing and accessing a large paritycheck matrix, encoding data, and analyzing code performance (e.g., determining a code’s distance properties). Recent research results [4–10] about LDPC codes with large girths show that there are many possible ways to construct the LDPC codes with large girths. However, some of the code constructions [4–10] are not satisfied for application due to the complicated constraints for the structures of paritycheck matrices. To solve the problem, this letter provides a different construction of LDPC codes with large girth. The column weight of our codes is 3 and the row weight is (). The BER performance of the proposed codes is near to those of Mackay’s random codes and Tanner’s QC codes.
This letter is structured as follows. In Section 2, we analyze the figures of 4cycles and 6cycles in the paritycheck matrix.
Section 3 presents the design algorithm of the paritycheck matrix. Section 4 provides three lemmas show that the proposed codes do not contain 4cycles and 6cycles. Section 5 evaluates the BER performance of the proposed codes for AWGN channels via computer simulations. Finally, In Section 6 we make some conclusions.
2. Definitions of the Cycles
An LDPC code has the paritycheck matrix of columns, where has ones in each column, ones in each row, and zeros elsewhere, then the paritycheck matrix has rows [1, 2]. A bipartite graph with check nodes and bit nodes can be created with the edges between the bit and check nodes if there are corresponding 1s in the paritycheck matrix . Such a graph is frequently called a Tanner graph [1, 11]. A cycle in a Tanner graph refers to a finite set of connected edges, the edge starts and ends at the same node, and it satisfies the condition that no node (except the initial and final node) appears more than once [4–10, 12]. The length of a cycle is simply the number of edges of the cycle.
A cycle also can be shown in a tree. Figure 1 shows a 4cycle in a tree, and Figure 2 shows a 6cycle in a tree. Figures 1 and 2 are not general descriptions for a 4cycle and 6cycle in a tree. It is also possible that a 4cycle or a 6cycle starts from a variable (bit) node, similar to the trees in Figures 1 and 2. In a tree, check nodes represent rows in paritycheck matrix, and variable (bit) nodes represent columns. We try to design a paritycheck matrix without 4cycles and 6cycles.
2.1. Figures of 4Cycles in ParityCheck Matrix
In Figure 1, two 1s (we mark them as and ) in row belong to column and , respectively, two 1s (we mark them as and ) in row belong to column and , respectively, so the figure of the 4cycle in the paritycheck matrix as shown in Figure 3. If there is no figure as shown in Figure 3 in paritycheck matrix, there is no 4cycle in the LDPC code. We denote 4cycles as .
2.2. Figures of 6Cycles in ParityCheck Matrix
Figure 4 shows that and in row belong to column and respectively, in column belongs to row , in column belongs to row , in row and in row belong to column , where , , , , , represent six 1s in 6cycle, as shown in Figure 4. We denote 6cycles as .
From Figure 4 we know that six 1s in a 6cycle belong to 3 rows and 3 columns equally, so we can get the number of 6cycles with different figures to be 6 from Figure 5.
Figure 5 shows six different kinds of figures of 6cycles. If there is no figure as shown in Figure 5 in the paritycheck matrix, there is no 6cycle in the LDPC code.
3. Design Algorithm of ParityCheck Matrix
If there is a paritycheck matrix , row weight is (), column weight is 3. We design three submatrices , , , combine them into matrix , , transpose into , expand into desired paritycheck matrix using identity matrices and cyclic shift matrices of the identity matrix randomly.
3.1. Design Algorithm of Submatrix
The design algorithm of is as following.
(1)Design a matrix with the dimension . , other elements are “0”(2)Let , , , where represents circularly shift in (1) for rightshifting steps circularly.(3)Let and its dimension is .
3.2. Design Algorithm of Submatrix
The design algorithm of is as following.
(1)Design a matrix with the dimension . , other elements are 0s(2)Let , the number of included in is (3)Let , , where represents circularly rightshifting for steps.(4)Let with the dimension .
3.3. Design Algorithm of Submatrix
The design algorithm of is as following.
(1)Designing a matrix with dimension . , other elements are “0.” For , we can get by the following way: (2)Let , , , where represents with circularly rightshifting steps.(3)Let .(4)Let there are copies of in so the dimension of is
We can get by the following way:
Let , then the dimension of is .
3.4. Expansion of
The expansion algorithm is as follows.
(1)Select an identity matrix with dimension ,(2)Let , , where represents the matrix with circularly rightshifting steps.(3)Exchanging the 1s in by the elements in the matrix set randomly and exchanging 0s by null matrices with the same dimension as , then we get the paritycheck matrix , whose dimension is .
From the structure of the paritycheck matrix , we can see that the proposed code is a deterministic LDPC code, so the proposed code has the advantage for encoding. To get the LDPC code with the large length (), we can expand the by selecting the desired dimension p of the identity matrix in (5) with stochastically shifting.
4. The Demonstration of the Code with Girth 8
In Section 3, we present a method for constructing paritycheck matrix with girth 8 based on the submatrices’ shifting. In this section, we add three lemmas with theoretic proofs showing that the paritycheck matrix does not contain 4cycles and 6cycles.
Lemma 1. If a matrix is constructed by (4), there is no 4cycle in .
Proof. We can know from Figure 3 that if there are 4cycles
in the matrix in (4), the edges connecting and must be in two different submatrices of , , because and are in the same row. The following formula
must be true:
In the
submatrix ,
the lengths and could be .
In the submatrix ,
the lengths and could be .
In the submatrix ,
the lengths and could be We can easily know that ,
so there is no 4cycle in the matrix .
Lemma 2. If a matrix is constructed by (4), there is no 6cycle in .
Proof. As shown in Figure 4, if there are 6cycles in the
matrix the edges connecting , and must be in , and respectively. Without loss of generality, we assume that is the longest length among the three lengths.
The following formula must be true:
In the
submatrix ,
the lengths , and could be .
In the submatrix ,
the length of , and could be In the submatrix ,
the lengths , and could be We can know that the edge of must be in ,
or else
We know that
and ,
so
Thus there is no 6cycle in the matrix ,
which means to be free from 6cycles.
We know that , so the matrix is also free of 4cycles and 6cycles.
Lemma 3. If there is no 4cycle and 6cycle in the matrix , there is no 4cycle and 6cycle in the paritycheck matrix .
Proof. If there is a 4cycle in the paritycheck matrix , from Figure 3 we can know that , , and must be in four different identity matrices or cyclic shift matrices of identity matrices. Since an identity matrix and a cyclic shift matrix of the in means a “1” in , there is a 4cycle in . However, we know that there is no 4cycle in , so there is no 4cycle in the paritycheck matrix . Using the same method, we can know that there is no 6cycle in the paritycheck matrix .
5. Performance Evaluation
To demonstrate the errorcorrecting performance, we constructed two rate1/2 LDPC codes by the proposed method. For the purpose of comparison, we also construct two classes of Tanner’s QC codes and Mackay’s random codes [10, 11, 13]. We get Mackay’s random codes from [13]. Tanner’s QC codes are constructed by the method introduced in Tanner’s paper [10, 11]. Both Mackay’s random codes and constructed Tanner’s QC codes for the comparison are good codes. The selected codes’ parameters are given in Table 1.

Table 1 lists the typical values of the row weight , the column weight , the code length , the code rate and the girth for the three codes. The girths of the LDPC codes in Table 1 are tested by the approach of [12]. Table 2 lists the typical values of the proposed codes for different high code and code lengths. In Tables 1 and 2, the means the proposed code with the code length n and m parity check bits, so do the and .

We simulate the proposed code’s errorcorrecting performance with the assumption that each code is modulated by BPSK and transmitted over additive white Gaussian noise (AWGN) channel. All the codes are decoded with the sumproduct algorithm [2, 3]. Figure 6 shows the simulated BER versus signaltonoise ratio (). As shown in Figure 6. The BER performance of the proposed codes is very close to MacKay's random codes and Tanner’s QC codes. From Figure 7, it can be seen that the proposed codes also have a good BER performance at high code rates 0.67, 0.7, and 0.75. The BER curves of the two codes of at high code rates 0.67, 0.7 are close since two LDPC codes with different code lengths may have approximate minimum weights [10]. In the expansion algorithm of the proposed code, we replace “1” in by the elements in the set randomly, which will lead to the minimum weights of the obtained codes to be different, the shorter codes may have larger minimum weights.
6. Conclusions
In this paper, we proposed a QC LDPC code without girth4 and girth6, three lemmas are provided to prove the short girths’ properties of the proposed codes. Simulation verified the good errorcorrecting performance of the proposed code, whose BER performance is near to those of Tanner’s QC codes and MacKay’s random codes [10, 11, 13]. The good BER performance can remain at high code rates.
Acknowledgments
This paper is supported by the National Natural Science Foundation of China under Grant no. 60572093, Specialized Research Fund for the Doctoral Program of Higher Education (20050004016) of China, and IITA Professorship Program of Gwangju Institute of Science and Technology, South Korea. Part of contents of this paper was presented on IET International Conference on Wireless Mobile and Multimedia Networks Proceedings (ICWMMN 2006).
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Copyright
Copyright © 2008 Jun Fan 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.