Unified Scaling of Polar Codes: Error Exponent, Scaling Exponent, Moderate Deviations, and Error Floors.

Consider the transmission of a polar code of block length N and rate R over a binary memoryless symmetric channel W and let be the block error probability under successive cancellation decoding. In this paper, we develop new bounds that characterize the relationship of the parameters R , N , , and the quality of the channel W quantified by its capacity I(W) and its Bhattacharyya parameter Z(W) . In previous work, two main regimes were studied. In the error exponent regime, the channel W and the rate scales roughly as 2^{-\sqrt {N}} . In the scaling exponent approach, the channel W and the error probability P\mathrm{ e} are fixed and it was proved that the gap to capacity I(W)-R scales as N^-1/\mu . Here, $\mu $ is called scaling exponent and this scaling exponent depends on the channel W$ . A heuristic computation for the binary erasure channel (BEC) gives \mu =3.627$ and it was shown that, for any channel W$ , 3.579 \le \mu \le 5.702$ . Our contributions are as follows. First, we provide the tighter upper bound \mu \le 4.714$ valid for any WN$ . In other words, we neither fix the gap to capacity (error exponent regime) nor the error probability (scaling exponent regime), but we do consider a moderate deviations regime in which we study how fast both quantities, as the functions of the block length N$ , simultaneously go to 0. Third, 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$ . Then, we vary the channel W scales as the Bhattacharyya parameter Z(W)$ raised to a power that scales roughly like {\sqrt {N}} . This agrees with the scaling in the error exponent regime. [ABSTRACT FROM PUBLISHER]