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

Abstract

Consider the transmission of a polar code of block length $N$ and rate $R$ over a binary memoryless symmetric channel $W$ and let $P_{\mathrm{ e}}$ be the block error probability under successive cancellation decoding. In this paper, we develop new bounds that characterize the relationship of the parameters $R$ , $N$ , $P_{\mathrm{ e}}$ , 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 $R are fixed, and it was proved that the error probability $P_{\mathrm{ e}}$ 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 $W$ . With the same technique, we obtain the upper bound $\mu \le 3.639$ for the case of the BEC; this upper bound approaches very closely the heuristically derived value for the scaling exponent of the erasure channel. Second, we develop a trade-off between the gap to capacity $I(W)-R$ and the error probability $P_{\mathrm{ e}}$ as the functions of the block length $N$ . 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 $W$ and study the impact of this variation on the error probability. We show that the error probability $P_{\mathrm{ e}}$ 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.