# 极化码构造原则

Arikan在论文[1]指出极化码构造原则：

Polar code construction rule. To construct an $\left( N,K \right)$ polar code for a B-DMC $W$, we select $A$ as the subset of indices $A\subset \left\{ 1,…,N \right\}$ such that $\left| A \right|=K$ and for each $i\in A$, the value $Z\left( W_{N}^{\left( i \right)} \right)$ is among the smallest $K$ values in the set $\left\{ Z\left( W_{N}^{\left( i \right)} \right):j=1,…,N \right\}$. The choice of the frozen vector $f_{1}^{\left| { {A}^{c}} \right|}$ is unspecified; any choice $\left( A,f_{1}^{\left| { {A}^{c}} \right|} \right)$ defines a polar code.

$Z\left( W \right)$与信道容量$I\left( W_{N}^{\left( i \right)} \right)$成反比，$I\left( W_{N}^{\left( i \right)} \right)$越大，$Z\left( W_{N}^{\left( i \right)} \right)$越小。信道极化的特性是，一部分子信道的容量趋于1，一部分子信道的容量趋于0。构造极化码时自然要选择容量较大的子信道来放置信息比特，也就是选择$Z\left( W_{N}^{\left( i \right)} \right)$最小的$K$个子信道。这就是Arikan定义的极化码构造原则。但问题是巴氏参数如何计算？

# 巴氏参数界

A pair of upper bounds on the Bhattacharyya parameters of bit-channels evolve as simply as $\left\{ z,z \right\}\to \left\{ 2z-{ {z}^{2}},{ {z}^{2}} \right\}$ at each polarizing transform $F$. Due to its simplicity, this construction has been widely used, and produced good polar codes.

# 高斯近似

An important characteristic of polar codes is their non-universality. That is, different polar codes are generated depending on the specified value of signal-to-noise ratio (SNR), known as the design-SNR. A change in operating SNR is possible in practice but a change in code according to SNR is not desirable. Therefore we wish to construct a polar code at one design-SNR and use it for a range of possible SNRs.

# design-snr

design-snr用于极化码的设计，它和实际信道的SNR不是一回事，而是“一对多”的关系。希望用一个design-snr适配于一定范围的SNR。design-snr有如下定义：

Algorithm PCC-0 : The Bhattacharyya bounds
Algorithm PCC-1 : The Monte-Carlo estimation
Algorithm PCC-2 : The full TPM estimation
Algorithm PCC-3 : The Gaussian approximation

design-snr不是通过计算得来的，而是直接给定。作者指出，最优design-snr是码率、码长和构造算法的函数。得到最优design-snr并不容易，作者是通过搜索算法进行大量的仿真试验，得到了这四种构造算法分别对应的最优design-snr：

PCC-0 @ 0dB
PCC-1 @ 1dB
PCC-2 @ -1.5917dB
PCC-3 @ -1.5917dB

# 总结

design-snr和高斯近似是一个比较好的极化码构造方案。你只需要考虑把${ {\sigma }^{2}}$构造为design-snr的函数，设定了一个design-snr，就意味着设定了${ {\sigma }^{2}}$。

design-snr已经是相当程度上的简化操作，让编码和译码器不必每时每刻都关注SNR，而只关注design-snr。

# 参考文献

[1] Arikan E. Channel polarization: A method for constructing capacity-achieving codes[C]//Information Theory, 2008. ISIT 2008. IEEE International Symposium on. IEEE, 2008: 1173-1177.
[2] Vangala H, Viterbo E, Hong Y. A comparative study of polar code constructions for the AWGN channel[J]. arXiv preprint arXiv:1501.02473, 2015.