Abstract
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 $Z(W)^{\sqrt{N}}$.