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 Pe 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, Pe, 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<I(W) are fixed, and it was proved that the error probability Pe scales roughly as 2-N. In the scaling exponent approach, the channel W and the error probability Pe are fixed and it was proved that the gap to capacity I(W)-R scales as N-1/μ. Here, μ is called scaling exponent and this scaling exponent depends on the channel W. A heuristic computation for the binary erasure channel (BEC) gives μ=3.627 and it was shown that, for any channel W, 3.579 μ 5.702. Our contributions are as follows. First, we provide the tighter upper bound μ 4.714 valid for any W. With the same technique, we obtain μ 3.639 for the case of the BEC, which approaches very closely its heuristically derived value. Second, we develop a trade-off between the gap to capacity I(W)-R and the error probability Pe as functions of the block length N. In other words, we consider a moderate deviations regime in which we study how fast both quantities, as 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 we show that the error probability Pe scales as the Bhattacharyya parameter Z(W) raised to a power that scales roughly like N.