Both solutions take advantage of the _nested_ property of polar codes for degraded channels, where the transmitter keeps sending additional code bits of a mother code to the receiver until a decoding success is announced. Then, the receiver can decode the whole block through a sequential decoding procedure. However, due to the block length and code-rate limitation of polar codes, both constructions of RC polar codes can achieve the capacity only for a sequence of rates that satisfy a certain relationship. Furthermore, these RC coding schemes can be used only when the family of channels over which the transmission takes place is ordered by degradation; otherwise, the nested property does not hold. Therefore, we develop an IR-HARQ transmission scheme based on a family of RC polar codes named parallel concatenated punctured polar (PCPP) codes in this paper. The PCPP coding scheme are optimized for a sequence of successively degraded channels and used for our IR-HARQ transmission scheme, for which the main ideas are exploited from concatenated polar coding construction that can be decoded by a sequence of parallel polar decoders [33]. The main contribution and novelty of this paper can be summarized as follows:  
* We investigate an improved random puncturing (IRP) pattern for the PCPP coding scheme to obtain a sequence of _nested_ encoding functions in a PCPP coding block transmission, for which the set of parity bits of a higher code-rate polar code is a subset of the set of parity bits of a lower code-rate polar code for a PCPP coding block IR-HARQ transmission. The proposed IRP algorithm only selects puncturing bits from the frozen bits set and keeps the information bits unchanged during puncturing, which can achieve 0.2-1 dB decoding performance improvement more than the existing random puncturing (RP) algorithm. * To realize multiple blocks transmission over a time-varying channel, we develop an RC IR-HARQ transmission scheme based on PCPP codes, where Refs. [32; 33] were only optimized in a single block transmission case. By analyzing the overhead of the previous successful decoded PCPP coding block in our IR-HARQ scheme, the optimal initial code-rate can be determined for each new PCPP coding block over time-varying channels. Simulation results show that the average number of transmissions is about 1.8 times for each PCPP coding block in our RC IR-HARQ scheme with a 2-level PCPP encoding construction, which reduces half of the average number of transmissions than the existing RC polar coding schemes. The remainder of this paper is organized as follows. In Section 2, we present some preliminaries of polar codes and rate-compatibility. In Section 3, we introduce our IRP algorithm that limits the selection of puncturing patterns on the set of frozen bits. Then, we propose our RC PCPP coding scheme for IR-HARQ transmission in Section 4. In Section 5, the optimal initial code-rate is analyzed for multiple RC PCPP coding blocks for IR-HARQ schemes over time-varying channels. The simulation results and comparisons for the proposed scheme and existing schemes are presented in Section 6. Finally, Section 7 concludes the paper with some remarks and suggestions of further research work.