## 6 Simulation and Comparison  
### Performance of the Improved Random Puncturing Algorithm  
In this subsection, we present Monte Carlo simulation results to evaluate the performance of our IRP algorithm over a binary-input additive white Gaussian noise (BI-AWGN) channel with antipodal signaling \(\pm\)**1**, and other system parameters are provided in Table 3.  
The bit error rate (BER) and frame error rate (FER) performance of our IRP algorithm, the random puncturing (RP) algorithm [25], the stop-tree puncturing (STP) algorithm [25], and the reliability-Type I algorithm [38] are shown in Figure 4. Obviously, we can observe that by avoiding information bit punctured, the IRP algorithm significantly outperforms the other three algorithms, and the STP algorithm provides the worst BER and FER performance due to the requirement of a BP decoding algorithm instead of the SCL decoding algorithm [8].  
When the punctured polar code-rate is \(\mathbf{R}_{j}=\mathbf{0.6}\), the performance of punctured codes under IRP algorithm obtains a 0.2 dB gain at BER \(\mathbf{10^{-4}}\) and 0.2 dB at FER \(\mathbf{10^{-2}}\) more than the RP algorithm.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Initial Code-Rate** & **Channel Capacity** & **Value Range of \(\mathbf{I^{(n+1)}}(W)\)** & **Transmission Rounds** & **Coding Pairs** \\ \hline \multirow{4}{*}{Case 1: \(R_{U}^{(n)}\)} & \multirow{2}{*}{increase} & \(\mathbf{I^{(n+1)}}(W)=R_{U}^{(n)}\) & 1 & 1 \\  & & \(\mathbf{I^{(n+1)}}(W)<R_{U}^{(n)}\) & 2 & 2 \\ \cline{2-5}  & \multirow{2}{*}{decrease} & \(R_{L}^{(n)}\leq I^{(n+1)}(W)\) & 2 & 2 \\ \hline \multirow{4}{*}{Case 2: \(R_{U}^{(n)}\)} & \multirow{2}{*}{increase} & \(\mathbf{I^{(n+1)}}(W)=R_{U}^{(n)}\) & 1 & 1 \\  & & \(\mathbf{I^{(n+1)}}(W)<R_{U}^{(n)}\) & 2 & 2 \\ \cline{2-5}  & \multirow{2}{*}{decrease} & \(R_{L}^{(n)}\leq I^{(n+1)}(W)\) & 2 & 2 \\  & & \(\mathbf{I^{(n+1)}}(W)<R_{L}^{(n)}\) & 3 & 3 \\ \hline \multirow{4}{*}{Case 3: \(R_{U}^{(n)}\)} & \multirow{2}{*}{increase} & \(\mathbf{I^{(n+1)}}(W)=I_{max}\) & 1 & 1 \\  & & \(\mathbf{R}_{U}^{(n)}\leq I^{(n+1)}(W)<I_{max}\) & 2 & 1 \\ \cline{1-1}  & & \(\mathbf{I^{(n+1)}}(W)<R_{U}^{(n)}\) & 3 & 2 \\ \cline{1-1} \cline{2-5}  & \multirow{2}{*}{decrease} & \(R_{U}^{(n)}\leq I^{(n+1)}(W)\) & 2 & 2 \\ \cline{1-1}  & & \(\mathbf{I^{(n+1)}}(W)<R_{U}^{(n)}\) & 3 & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The relationship between the optimal initial code-rate of the PCPP coding block and two key parameters.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Block length of mother polar codes \(\mathbf{N_{k}}\) & 1024 \\ Mother polar code-rate \(\mathbf{R_{k}}\) & 0.5 \\ Punctured polar code-rates \(\mathbf{R_{j}}\) & 0.6, 0.7, 0.8 \\ Polar codes decoder & SCL \\ Modulation & BPSK \\ \hline SCL: SCL decoding; BPSK: Binary Phase Shift Keying.  
\end{table}
Table 3: Simulation parameters of the puncturing algorithms.  
With the increasing of the punctured polar code-rate, the advantages of our proposed IRP algorithm increases. When the punctured polar code-rate is \(\mathbf{R_{p}}=\mathbf{0.7}\), the IRP algorithm outperforms the RP algorithm and obtains a 1 dB gain at BER \(\mathbf{10^{-4}}\) and 0.3 dB at FER \(\mathbf{10^{-2}}\), respectively. Moreover, the RP algorithm suffers from an error-floor when the punctured polar code-rate is \(\mathbf{R_{p}}=\mathbf{0.8}\), where our IRP algorithm can completely avoid the problem.