## 5 Rate-Compatible IR-HARQ Transmission Scheme Based on PCPP Codes
### Optimization Method for the 1 st PCPP Coding Block of the RC IR-HARQ Transmission Scheme  
As analyzed above, an estimation interval of the channel capacity can be obtained after each data block is transmitted successfully in our RC IR-HARQ transmission scheme. Here, let \([R_{k}^{1},R_{k-1}^{1})\) denote the capacity interval after the 1 st block is transmitted successfully and \(I^{(1)}(W)\in[R_{k}^{1},R_{k-1}^{1})\). Then, the initial code-rate of \(C_{2}(N_{1},R_{1},\mathcal{A}_{1}^{(1)})\) for the 2 nd data block can be chosen according to \([R_{k}^{1},R_{k-1}^{1})\). Furthermore, let \([R_{L}^{(n)},R_{U}^{(n)})\) denote the capacity interval after the \(n\)-th data block is transmitted successfully, and \(I^{(n)}(W)\) denote the corresponding channel capacity and \(I^{(n)}(W)\in[R_{L}^{(n)},R_{U}^{(n)})\). In a practical time-varying wireless system, the estimation error of the real channel state information is hard to eliminate. However, our optimization method for the 1 st PCPP coding block of RC IR-HARQ transmission scheme can work well, and we will discuss the affection of the estimation error in the following. First, let \([I_{min},I_{max}]\) denote the variation range of the time-varying channel capacity that can be obtained from the statistical information of the time-varying channel; \(\Delta R\) denotes the difference of the code-rates between any two adjacent PCPP coding blocks; \(C_{n}(N_{k},R_{k},\mathcal{A}_{k}^{(k)})\) is the \(k\)-th PCPP coding block in the \(n\)-th block transmission, where the receiver begins the backward decoding with the \(k\)-th received PCPP coding block; \(C_{n}([N_{k}]_{p},R_{j},\mathcal{A}_{j}^{(k)},\mathbf{P}_{j}^{N_{k}})_{j=k}^ {I}\) is a sequence of nested punctured polar codes, where \(\mathbf{p}_{j}^{N_{k}}=(p_{1},p_{2},...,p_{N_{k}})\in\{0,1\}^{N_{k}}\) is the \(j\)-th puncturing pattern and the mother code is \(C_{n}(N_{k},R_{k},\mathcal{A}_{k}^{(k)})\). Recall the variation range of the time-varying channel capacity \([I_{min},I_{max}]\). Then, our optimization method for choosing the initial code-rate of the \((n+1)\)-th data block \(C_{n+1}(N_{1},R_{1},\mathcal{A}_{1}^{(1)})\) from the capacity interval \([R_{L}^{(n)},R_{U}^{(n)})\) can be distinguished into three possible cases, which are illustrated in Figure 3.  
**Case 1:** the minimum channel capacity satisfies \(\mathbf{R}_{L}^{(n)}\leq\mathbf{I}_{min}\) and the maximum channel capacity satisfies \(\mathbf{I}_{max}\leq\mathbf{R}_{U}^{(n)}\) as shown in Figure 3a. Then, the optimal initial code-rate for the \((n+1)\)-th block is \(\mathbf{R}_{U}^{(n)}\). If the time-varying channel capacity increases, and the channel capacity satisfies \(\mathbf{I}^{(n)}(W)\leq\mathbf{I}^{(n+1)}(W)<\mathbf{I}_{max}\), since the maximum channel capacity is \(\mathbf{I}_{max}\leq\mathbf{R}_{U}^{(n)}\), and the initial code-rate is \(\mathbf{R}_{1}^{(n+1)}=\mathbf{R}_{U}^{(n)}\), which is beyond the channel capacity as \(\mathbf{I}_{max}<\mathbf{R}_{U}^{(n)}\), then the rate of the 2 nd PCPP coding block is \(\mathbf{R}_{2}^{(n+1)}=\mathbf{R}_{L}^{(n)}\). Hence, a 2-level PCPP code for the \((n+1)\)-th block can be constructed with rates \(\mathbf{R}_{1}^{(n+1)}=\mathbf{R}_{U}^{(n)}\) and \(\mathbf{R}_{2}^{(n+1)}=\mathbf{R}_{L}^{(n)}\), and two time transmissions can ensure the data block delivery, and we have \(\mathbf{I}^{(n+1)}(W)\in[\mathbf{R}_{L}^{(n)},\mathbf{R}_{U}^{(n)})\) at the end of this data block transmission. In particular, if \(\mathbf{I}^{(n+1)}(W)=\mathbf{I}_{max}=\mathbf{R}_{U}^{(n)}\), the \((n+1)\)-the data block with the initial code-rate \(\mathbf{R}_{1}^{(n+1)}=\mathbf{R}_{U}^{(n)}\) PCPP coding block can be transmitted successfully in one time transmission. Else, if the time-varying channel capacity decreases, the channel capacity satisfies \(\mathbf{I}_{min}\leq\mathbf{I}^{(n+1)}(W)<\mathbf{I}^{(n)}(W)\), and the initial code-rate is \(\mathbf{R}_{1}^{(n+1)}=\mathbf{R}_{U}^{(n)}>\mathbf{I}_{max}\).