## 2 Preliminaries
### Rate-Compatible Polar Codes  
Let \(\{W_{1},W_{2},...,W_{J}\}\) denote a family of \(J\) channels. If their respective capacity is \(I(W_{1})>I(W_{2})>...>I(W_{J})\), we call that a sequence of successively _degraded channels_[37] and it is referred as \(I(W_{1})\succeq I(W_{2})\succeq...\succeq I(W_{J})\). A family of _rate-compatible_ (RC) polar codes \(\{C(N,R_{i},\mathcal{A}_{i})\}|_{i\in\{1,2,...,J\}}\) can be designed for such degraded channels where their respective information sets are such that \(\mathcal{A}_{1}\supseteq\mathcal{A}_{2}\supseteq...\supseteq\mathcal{A}_{J}\), which is called _nested_ property. Each polar code for \(W_{i}\) has the same block length \(N\) and with code-rates \(R_{1}>R_{2}>...>R_{J}\) and each \(R_{i}=I(W_{i})\).  
[PERSON] et al. [32] and [PERSON] et al. [33] proposed two RC polar-like coding schemes. In both schemes, the sender first transmits the code block \(C(N,R_{1},\mathcal{A}_{1})\) with a predetermined maximum rate \(R_{1}\). If the decoding is successful at the receiver, this block is successfully transmitted and the procedure ends; otherwise, an error message (NACK) is fed back to the sender, and the received data is stored in the buffer and waits for decoding with further transmission data blocks. In the second transmission, the code block \(C(N,R_{2},\mathcal{A}_{2})\) with rate \(R_{2}=R_{1}/2\) will be transmitted. The second code block is constructed by the information bits that were put on the indices \(\mathcal{A}_{1}\setminus\mathcal{A}_{2}\) in the first transmission, and these information bits are now put on the indices \(\mathcal{A}_{2}\), and rest are frozen bits. Here, \(\mathcal{A}_{1}\setminus\mathcal{A}_{2}\) represents the relative complement of \(\mathcal{A}_{2}\) in \(\mathcal{A}_{1}\).  
The transmission will be stopped until the \(k\)-th code block \(C(N,R_{k},\mathcal{A}_{k})\) is decoded successfully, where \(R_{k}=R_{1}/k\) and the information bits are those with the indices \(\mathcal{A}_{(k-1)}\setminus\mathcal{A}_{k}\) in all of the previous transmissions, and the length of the frozen bits is \(N-|\mathcal{A}_{k}^{c}|\).  
After the \(k\)-th code block \(C(N,R_{k},\mathcal{A}_{k})\) is decoded successfully, then the receiver can decode the previous buffered code block in a backward decoding strategy according to the nested property. The decoded bits \(\mathcal{A}_{k}\) will be used as frozen bits to help decode the \((k-1)\)-th polar code block \(C(N,R_{k-1},\mathcal{A}_{k-1})\). Then, the code-rate of the \((k-1)\)-th code block is reduced to \(R_{k}\). Therefore, the \(k-1\)-th code block can be decoded. Likewise, all the previous buffed \((k-1)\) code blocks can be decoded backwards after \((k-1)\) times decoding and all the \(\mathcal{A}_{1}\) information bits are recovered.  
Though these schemes can provide a simple way to construct a family of RC codes with the degraded rates \(R,R/2,R/3,...\), we find it is unable to achieve arbitrary rates in [32], and the puncturing scheme in [25] caused serious performance loss, and the information bits may be punctured and the nested property does not hold. Moreover, both of the schemes lack introduction of how to measure the reliabilities of the subchannels without any channel information in the first transmission. Thus, we need to address these problems in our new scheme in this paper.