## 4 System Model  
Base on a family of the above RC punctured polar codes, we can construct our parallel concatenated punctured polar (PCPP) codes \(\{C(N_{i},R_{i},\mathcal{A}_{i}^{(i)})\}_{|i\in\{1,2,...,j\}}\) over any sequence of successively degraded channels \(W_{1}\succeq W_{2}\succeq...\succeq W_{J}\) to transmit \(K\) information bits. Their information sets \(\{\mathcal{A}_{i}^{(i)}\}\) are nested and the block length \(N_{i}\) and the corresponding code-rate \(R_{i}\) for each PCPP coding block are satisfied  
\[R_{i}=\frac{K}{\sum_{j=1}^{i}N_{j}}, \tag{7}\]  
where \(K=N_{1}R_{1}\) is equal to the number of the information bits in the PCPP codes. In addition, for each block length \(N_{i}\), we can use our IRP algorithm to construct a sequence of nested punctured polar codes \(\{C([N_{i}]_{p},R_{j},\mathcal{A}_{j}^{(i)})\}_{j=i}^{J}\) with rates \(\{R_{i},R_{i+1},...,R_{J}\}\), and the length of the information bits in each punctured code block is the same to satisfy  
\[\mathcal{A}_{j}^{(i)}=[N_{i}]_{p}R_{j}. \tag{8}\]  
The RC PCPP coding scheme for IR-HARQ transmission is described as follows. In the first transmission, the sender transmits the code block \(C(N_{1},R_{1},\mathcal{A}_{1}^{(1)})\) while carrying all the \(K\) information bits. If the first transmission fails, the received data will be stored and the second code block \(C(N_{2},R_{2},\mathcal{A}_{2}^{(2)})\) will be transmitted. The information bits located at the indices \(\mathcal{A}_{1}^{(1)}\setminus\mathcal{A}_{2}^{(1)}\) of the first block should be put at the current information set \(\mathcal{A}_{2}^{(2)}\) and encoded, where \(\mathcal{A}_{j}^{(i)}\) is the information set that consists of the \(N_{i}R_{j}\) most reliable indices when the block length is \(N_{i}\). If the second transmission also fails, the third block \(C(N_{3},R_{3},\mathcal{A}_{3}^{(3)})\) will be constructed by the information bits located at the indices \(\mathcal{A}_{2}^{(1)}\setminus\mathcal{A}_{3}^{(1)}\) of the first block and the information bits located at indices \(\mathcal{A}_{2}^{(2)}\setminus\mathcal{A}_{3}^{(2)}\) of the second block. Continuously, the transmission will be stopped if the decoding succeeds after the \(k\)-th block  
Figure 1: An example of puncturing from a generation matrix, where \(N_{M}=8\), and \([N_{M}]_{p}=6\). \(C(N,R_{k},\mathcal{A}_{k}^{(k)})\) is transmitted or \(k=J\). Then, we could decode the previous blocks backward to recover all of the \(K\) information bits as well as the decoding of the RC polar codes in Section 2.3. In the following, we will give a more detailed explanation of the main idea through a 3-level RC PCPP coding scheme as shown in Figure 2. In this case, a information set of 12 bits will be transmitted through three transmissions and a 3-level PCPP code block is needed to encode where with the parameters \(N_{1}=16\), \(R_{1}=3/4\), \(R_{2}=1/2\), \(R_{3}=1/4\). We start by constructing the PCPP coding block \(C(N_{1},R_{1},\mathcal{A}_{1}^{(1)})\) for the 1 st transmission, as shown in Figure 2a, where \(u^{12}=\{u_{1},u_{2},...,u_{12}\}\) are the information bits. When the 1 st transmission fails, the 2 nd PCPP coding block \(C(N_{2},R_{2},\mathcal{A}_{2}^{(2)})\) will be sent, where \(N_{2}=8\) can be calculated according to Equation (7), and the information bits are \(u^{4}=\{u_{1},u_{2},u_{3},u_{4}\}\) by calculating \(\mathcal{A}_{1}^{(1)}\setminus\mathcal{A}_{2}^{(1)}\). Assuming the 2 nd transmission also fails in this example, the sender performs the 3 rd transmission with \(N_{3}=24\), for which the block length is not satisfy power of two and can not be obtained by polar encoder directly. Then, we use the IRP algorithm to construct the 3 rd PCPP coding block. In addition, to avoid the block error caused by puncturing too many bits, we should do our best to minimize the number of puncturing bits.