## 6 Simulations  
The simulation results of the proposed blind recognition algorithm are shown in this section. In the simulations, we assume that the propagation channel is a Binary Symmetry Channel (BSC) which is corrupted by an AWGN with the variance \(\sigma_{n}^{2}=N_{0}/2\). For each configuration, the information symbols in the codes are randomly chosen and the modulation mode is BPSK. All the simulations have the same settings of the observation window with length \(L=3\),000 bits and the searching range of the code length is 7-127.  
When applying the algorithm to BCH(63,45) codes, the simulation results of the code length estimation are shown in Figures 4-6. The signal is corrupted by an AWGN with the SNR E\({}_{y}\)/N\({}_{0}=5\) dB.  
Figure 3: Recognizing the BCH(63, 51) codes using a primitive polynomial different from that of the encoder.  
In Figure 4, we draw the stems of \(p_{i}\) on different elements \(\alpha^{\prime}\) taken from GF(\(2^{m}\)) when the code length and synchronization positions are correct. It is shown in Figure 4 that the values of \(p_{i}\) on the code roots are obviously higher than those on the other elements. And we also draw the stems on a fault code length and synchronization position in Figure 5.  
Comparing with Figure 4, the values of \(p_{i}\) in Figure 5 is uniform, so the information entropy of \(p_{i}\) on correct coding parameters should be lower and the corresponding dispersion entropy function is higher. As shown in Figure 6, we draw the stems of RIDEF on different code length \(l\) and coding starting positions \(t\) when the start position \(t=0\) of the observation window is at the fortieth bit of a codeword. In the figure, the value of \(p_{i}\) in the case of \(l=63\) and \(t=23\) is the highest, thus we consider the parameters \(l=63\) and \(t=23k\left(k\in\mathbb{Z}\right)\) as the estimated code length and synchronization positions. The result is in accordance with the simulation settings.  
Figure 4: Values of \(p_{i}\) on correct code length and synchronization positions.  
Figure 5: Values of \(p_{i}\) on incorrect code length and synchronization positions.  
The performance of the algorithm is affected by the channel quality. In Figure 7, we draw the performance of the proposed algorithm when applied to code length and synchronization positions recognitions for several different binary BCH codes, including the shortened codes. The curves depict the false recognition probabilities (FRP) of the code length and coding starting positions estimations on different SNRs. We also compare the performance of our proposed recognition algorithm with the hard-decision-based RIDERS algorithm proposed in [22; 23; 24]. The PFR of our proposed algorithm fall rapidly when SNR increases, and it is much lower than that of the previous hard decision algorithms on each single SNR value.  
After the code length and synchronization position estimation, the generator polynomial can be recognized by searching for the code roots according to the steps proposed in Section 3. Figure 8 shows the performance of the proposed generator polynomial recognition algorithm when applied to several different binary BCH codes, which are BCH(63, 51), BCH(63, 39) and BCH(30, 20). The curves show the false recognition probabilities on different noise levels. As E\({}_{\text{s}}\)/N\({}_{0}\) rises, the curves fall rapidly. When E\({}_{\text{s}}\)/N\({}_{0}\) is above 5 dB, no false recognition occurred during our 200,000 instances of simulation. We also compare our proposed algorithm with the previous hard-decision-based recognition algorithms proposed by [22; 23; 24] in the figure. It shows that the recognition performance is improved obviously in soft decision situations. A gap of 1-2 dB exists between the two groups of curves.  
Figure 6: Code length and synchronization positions estimation of BCH(63, 45) codes.