## 6 Simulation Results
### Performance on Lossless Source-Relay and Relay-Destination Links  
We now consider the simple case where no symbols are erased on each source-relay and relay-destination link. Figure 5 shows the theoretical asymptotic bit error rate (BER) performance before and after the optimization when the source number \(n=3\). Note that the theoretical BER of source \(S_{2}\) and \(S_{3}\) are identical since they have the same connection and the same degree distribution. As shown in Figure 5, it indicates that after the optimization, the BER performance is improved significantly, which verifies the effectiveness of the optimization. For example, when the overall overhead \(\epsilon=0.2\), the BER of \(S_{1}\) is approximately \(1/3\) of that before optimization and the BER of \(S_{2}\) and \(S_{3}\) is approximately \(1/20\) of that before optimization. Furthermore, we compare the theoretical BER with the simulated finite-length BER of \(k=100\) using optimized degree distributions for \(n=3\) in Table 1. It reveals that the simulated BER is consistent with asymptotic BER as the code overhead increases, which verifies the correctness of the optimization method in Section 5. In addition, Figure 5 shows that the BER performance of sources connected to the cooperative relay is improved, whereby the UEP property is obtained by EDFC.  
Figures 6 and 7 compare the performance of EDFC with GDFC [5] in two-source and four-source networks, respectively. The optimized degree distributions in Table 1 are both used for simulations of these two coding schemes. In a four-source network, source \(S_{1}\) and \(S_{2}\) have similar performances, as do \(S_{3}\) and \(S_{4}\). Therefore, for simplicity, we only plot the simulated BER curves for \(S_{1}\) and \(S_{3}\) in Figure 7. It reveals that EDFC outperforms GDFC both in two-source and four-source networks, and the performance benefit is not restricted to the sources connected to the cooperative relay but applies to all sources in the network. For example, in Figure 6, when the overall overhead \(\epsilon=0.3\), the BER of \(S_{1}\) for EDFC is approximately \(1/25\) of that for GDFC, and the BER of \(S_{2}\) for EDFC is approximately \(1/150\) of that for GDFC. The reason is that EDFC has an additional cooperative relay, which enhances the performance of the source connected to it. Moreover, as a result of the involvement of the cooperative relay, the number of output symbols with degree 1 grows in the Tanner graph, which can increase the decoding success rate of the Tanner graph. Thus, an improvement of the overall performance is obtained.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \multirow{2}{*}{**Degree**} & \multicolumn{4}{c}{_n = 2, \(t=1\)_} & \multicolumn{4}{c}{_n = 3, \(t=2\)_} & \multicolumn{4}{c}{_n = 4, \(t=2\)_} \\ \cline{2-13}  & \(\Phi_{1}(x_{1})\) & \(\Phi_{2}(x_{2})\) & \(\Gamma_{1}(x)\) & \(\Gamma_{2}(x)\) & \(\Phi_{1}(x_{1})\) & \(\Phi_{2}(x_{2})\), & \(\Phi_{3}(x_{3})\) & \(\Gamma_{1}(x)\) & \(\Gamma_{2}(x)\) & \(\Phi_{1}(x_{1})\), \(\Phi_{3}(x_{3})\), & \(\Gamma_{1}(x)\) & \(\Gamma_{2}(x)\) \\ \hline
1 & 0.16 & 0.16 & 0.6654 & 1 & 0.13 & 0.13 & 0.6899 & 0.79 & 0.22 & 0.22 & 0.8579 & 0.68 \\
2 & 0.4386 & 0.1424 & 0.3346 & - & 0.4134 & 0.3273 & 0.23 & 0.21 & 0.0272 & 0.1698 & 0.12 & 0.32 \\
3 & 0.2845 & 0.5418 & - & - & 0.3626 & 0.4249 & 0.0801 & - & 0.6235 & 0.4966 & 0.01 & - \\
13 & 0.01 & \(<\)0.005 & - & - & \(<\)0.005 & \(<\)0.005 & - & - & \(<\)0.005 & \(<\)0.005 & 0.0121 & - \\
14 & 0.1069 & 0.1305 & - & - & \(<\)0.005 & 0.0109 &  
Figure 5: The comparison of BER performance before and after optimization for EDFC in a three-source network.  
Figure 6: BER performance comparison of EDFC and GDFC for two-source networks.