# Minimum Distance Optimization with Chord Edge Growth for High Girth Non-Binary LDPC Codes
## 4 Complexity Analysis and Design Examples
### Design Examples  
Several examples of finite-length NB-LDPC codes constructed using the proposed method are presented. All of the codewords are simulated on a binary input additive white Gaussian noise (AWGN) channel while using binary phase shift keying (BPSK) modulation, and decoded by the fast Fourier transform belief propagation (FFT-BP) algorithm [6]. The maximum number of iterations is set to be 50. In addition, all of the simulations terminated with 50 erroneous codewords (frames). **Example 1**. : _Based on the Hamiltonian graph with 26 vertices that are shown in Figure 2, the NB-LDPC codes with a code length of 312 bits and a code rate of 1/2 defined on GF(64) are constructed. The local minimum Hamming distance that is associated with a check node is used as a criterion to determine the set of non-zero entries for each row. One set of non-zero entries over GF(64) provided by [PERSON] et al. [15] is applied to the proposed method. The non-zero entry set \(R=\{\alpha^{37},\alpha^{22},\alpha^{9},\alpha^{0}\}\), with a minimum distance of \(3\) and a weight enumerator coefficient of \(20\), is selected. Subsequently, the proposed method is performed to configure the non-zero entries for the parity-check matrix. The proposed method and a simplified version of the proposed method are, respectively, used to construct NB-LDPC codes. These two versions are called PCEG and S-PCEG, respectively. For the design example using the S-PCEG method, after completing the search for the non-zero entries of the sixth chord edge, one candidate configuration is reserved. In addition, the proposed method is applied to the optimization of the NB-LDPC code with an efficient encoding structure that was proposed by [PERSON] et al. [24]. Based on the same basic graph and non-zero elements, NB-LDPC codes are randomly constructed for comparison._  
Table 2 shows the cycle distributions of the constructed codes. The girths of these NB-LDPC codes are all 12, and the numbers of cycles with lengths from 12 to 20 that do not satisfy FRC are listed. Using the proposed method, the cycles less than 20 in length are cancelled, which improves the performance to a certain extent. For cycles with a length of 20, the number of cycles not meeting FRC is further reduced. The similar effect of cycle cancellation is obtained in the parity-check matrix with an efficient encoding structure [24]. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Methods** & **The Total Number of Candidate Configurations** & **The Total Number of Related Cycles** & **The Total Number of FRC Checks** \\ \hline Progressive chord edge growth & 676,598 & 2330 & \(\approx\)1.32 \(\times\) 10\({}^{7}\) \\ \hline Iterative and row-wise optimization & 967,680 & 11,188 & \(\approx\)4.51 \(\times\) 10\({}^{8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The complexity of the optimization for the NB-LDPC code with the code rate of 3/4. Figures 8 and 9 shows the performance simulation results of different constructed codes. Because there is no short cycle not satisfying the FRC, the codes constructed by the proposed method have performance advantages in the high signal-to-noise ratio (SNR) region when compared with the codes constructed randomly with the same code parameters. The proposed method has the potential to construct NB-LDPC codes that meet the requirements of low bit error rate (BER) or frame error rate (FER). Applying the proposed method to the NB-LDPC code with an efficient encoding structure proposed by [PERSON] et al. (2014), an NB-LDPC code with low encoding complexity and superior performance can be constructed. Furthermore, the performance of the code constructed while using the proposed method under different decoding iterative number is analyzed, as shown in Figure 10. It can be observed that, when the iteration  
**Example 2**.