An Information-Theoretic Perspective on Successive Cancellation List Decoding and Polar Code Design
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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∈{8,32,128}, outperforming 5G designs.
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