An Information-Theoretic Perspective on Successive Cancellation List Decoding and Polar Code Design

This work identifies information-theoretic quantities that are closely related to the required list size for successive cancellation list (SCL) decoding to implement maximum-likelihood decoding. It also provides an approximation for these quantities that can be computed efficiently for very long codes. There is a concentration around the mean of the logarithm of the required list size for sufficiently large block lengths. We further provide a simple method to estimate the mean via density evolution for the binary erasure channel (BEC). Simulation results for the binary-input additive white Gaussian noise channel as well as the BEC demonstrate the accuracy of the mean estimate. A modified Reed-Muller code with dynamic frozen bits performs very close to the random coding union (RCU) bound down to the block error rate of $10^{-5}$ under SCL decoding with list size $L=128$ when the block length is $N=128$. The analysis shows how to modify the design to improve the performance when a more practical list size, e.g., $L=32$, is adopted while keeping the performance with $L=128$ unchanged. For the block length of $N=512$, a design performing within $0.4$ dB from the RCU bound down to the block error rate of $10^{-6}$ under an SCL decoder with list size $L=1024$ is provided. The design is modified using the new guidelines so that the performance improves with practical list sizes, e.g., $L\in\{8,32,128\}$, outperforming 5G designs.