Unified scaling of polar codes: Error exponent, scaling exponent, moderate deviations, and error floors

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_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_e scales roughly as 2^-√(N). In the scaling exponent regime, the channel W and P_e are fixed, while the gap to capacity I(W) - R scales as N^-1/μ, with 3.579 ≤ μ ≤ 5.702 for any W. We develop a unified framework to characterize the relationship between R, N, P_e, and W. First, we provide the tighter upper bound μ ≤ 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 we study how fast both the gap to capacity I(W) - R and the error probability P_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_e scales roughly as Z(W)^√(N).