## 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
Similarly, we then need to construct the 2 nd PCPP coding block with rate \(R_{2}^{(n+1)}=R_{L}^{(n)}\), and this can be decoded by receiver. Therefore, a 2-level PCPP codes for the \((n+1)\)-th data block is needed to construct with rates \(R_{1}^{(n+1)}=R_{U}^{(n)}\) and \(R_{2}^{(n+1)}=R_{L}^{(n)}\), and the final state is \(I^{(n+1)}(W)\in[R_{L}^{(n)},R_{U}^{(n)})\). **Case 2:** if the minimum channel capacity satisfies \(R_{L}^{(n)}-\Delta R\leq I_{min}<R_{L}^{(n)}\) and the maximum channel capacity satisfies \(I_{max}\leq R_{U}^{(n)}\) as shown in Figure 3b, then, the optimal initial code-rate for the \((n+1)\)-th block is \(R_{U}^{(n)}\). If the time-varying channel capacity increases, and the channel capacity satisfies \(I^{(n)}(W)\leq I^{(n+1)}(W)<I_{max}\) with the maximum channel capacity satisfying \(I_{max}\leq R_{U}^{(n)}\), similar to the previous case, a 2-level PCPP code for the \((n+1)\)-th data block needs to be constructed with rates \(R_{1}^{(n+1)}=R_{U}^{(n)}\) and \(R_{2}^{(n+1)}=R_{L}^{(n)}\) to successfully deliver the data block in two time transmissions with \(I^{(n+1)}(W)\in[R_{L}^{(n)},R_{U}^{(n)})\) for the next data block transmission. In addition, if \(I^{(n+1)}(W)=I_{max}=R_{U}^{(n)}\) and the initial code-rate for the \((n+1)\)-th data block is \(R_{1}^{(n+1)}\), the data block can be delivered in one time transmission. Else, if the channel capacity decreases, we can distinguish two subcases: if the channel capacity satisfies \(R_{L}^{(n)}\leq I^{(n+1)}(W)<I^{(n)}(W)<I_{max}\), the construction of the \((n+1)\)-th data block is the same as in **Case 1**, which is a 2-level PCPP code with rates \(R_{1}^{(n+1)}=R_{U}^{(n)}\) and \(R_{2}^{(n+1)}=R_{L}^{(n)}\); if the channel condition goes through a severe deterioration as \(R_{L}^{(n)}-\Delta R\leq I^{(n+1)}(W)<I^{(n)}(W)<R_{L}^{(n)}\), the initial code-rate \(R_{1}^{(n+1)}=R_{U}^{(n)}\) and 2 nd code-rate \(R_{2}^{(n+1)}=R_{L}^{(n)}\) are higher than the channel capacity, until the 3 rd PCPP coding block with the code-rate \(R_{3}^{(n+1)}=R_{L}^{(n)}-\Delta R\) is transmitted, and the receiver can begin the backward decoding. Therefore, a 3-level PCPP codes for the \((n+1)\)-th data block needs to be constructed, and \(I^{(n+1)}(W)\in[R_{L}^{(n)}-\Delta R,R_{L}^{(n)})\) is fed back to the sender. **Case 3:** if the minimum channel capacity satisfies \(R_{L}^{(n)}\leq I_{min}\) and the maximum channel capacity satisfies \(R_{U}^{(n)}\leq I_{max}\leq R_{U}^{(n)}+\Delta R\) as shown in Figure 3c, then the optimal initial code-rate for the \((n+1)\)-th block is \(I_{max}\), which is obtained by puncturing a mother code \(C_{n+1}(N_{L},R_{U},\mathcal{A}_{1}^{(1)})\). In this case, when the channel capacity increases, the channel capacity can increase to \(R_{U}^{(n)}\leq I^{(n+1)}(W)\leq I_{max}\leq R_{U}^{(n)}+\Delta R\). If we choose \(R_{U}^{(n)}\) as the initial code-rate for the \((n+1)\)-th block, the \((n+1)\)-th data block can be transmitted successfully in one time transmission. However, if we do so, we will have \(R_{U}^{(n+1)}=R_{U}^{(n)}\), which is decreasing the efficiency of the channel use. Therefore, the sender in Case 3 needs to construct a punctured polar code \(C_{n+1}(N_{L},I_{max},\mathcal{A}_{1}^{(1)})\) with initial code-rate \(R_{1}^{(n+1)}=I_{max}\) from the mother code \(C_{n}\big{(}N_{L},R_{U},\mathcal{A}_{k}^{(k)}\big{)}\) with code-rate \(R_{k}^{(n)}=R_{U}\). If the channel capacity increases, we can distinguish three subcases: If \(I^{(n+1)}(W)=I_{max}\), the \((n+1)\)-th data block with the initial code-rate \(R_{1}^{(n+1)}=I_{max}\) PCPP coding block can be transmitted successfully in one time transmission, and the feedback channel capacity interval is \(I^{(n+1)}(W)\in[I_{max}-\Delta R,I_{max})\). Else, if \(R_{U}^{(n)}\leq I^{(n+1)}(W)<I_{max}\), a 2-level PCPP code with rates \(R_{1}^{(n+1)}=I_{max}\) and \(R_{2}^{(n+1)}=R_{U}^{(n)}\) can transmit the \((n+1)\)-th data block successfully in two time transmissions with \(I^{(n+1)}(W)\in[R_{L}^{(n)},R_{U}^{(n)})\) for the next data block transmission.