Unified Scaling of Polar Codes: Error Exponent, Scaling Exponent, Moderate Deviations, and Error Floors
Marco Mondelli Ruediger Urbanke Hamed Hassani
Proceedings of the 2015 IEEE International Symposium on Information Theory, Hong Kong, China, June 2015

Consider transmission of a polar code of block length $N$ and rate $R$ over a binary memoryless symmetric channel $W$ with capacity $I(W)$ and Bhattacharyya parameter $Z(W)$ and let $P_{\mathrm{e}}$ be the error probability under successive cancellation decoding. Recall that in the error exponent regime, the channel $W$ and $R < I(W)$ are fixed, while $P_{\mathrm{e}}$ scales roughly as $2^{-\sqrt{N}}$.  In the  scaling exponent regime, the channel $W$ and $P_{\mathrm{e}}$ are fixed, while the gap to capacity $I(W) - R$ scales as $N^{-1/\mu}$, with $3.579 \le mu \le 5.702$ for any $W$.  We develop a unified framework to characterize the relationship between $R$, $N$, $P_{\mathrm{e}}$, and $W$.  First, we provide the tighter upper bound $\mu \le 4.714$, valid for any $W$.  Furthermore, when $W$ is a binary erasure channel, we obtain an upper bound approaching very closely the value which was previously derived in a heuristic manner.  Secondly, we consider a  moderate deviations regime and study how fast both the gap to capacity $I(W) - R$ and the error probability $P_{\mathrm{e}}$ simultaneously go to $0$ as $N$ goes large. Thirdly, we prove that polar codes are not affected by  error floors . To do so, we fix a polar code of block length $N$ and rate $R$, we let the channel $W$ vary, and we show that $P_{\mathrm{e}}$ scales roughly as $Z(W)^{\sqrt{N}}$.