A New Metric Function for SC-based Polar Decoders: Polarization, Pruning, and Fast Decoders

In this paper, we propose a method to obtain the optimal metric function at each depth of the polarization tree through a process we call polarization of the metric function. This polarization process generates an optimal metric at intermediate levels of the polarization tree, which can be applied in fast successive-cancellation-based (FSC) and SC list-based (FSCL) decoders -- decoders that partially explore the binary tree representation. We prove that at each step of the polarization tree, the expected value of the metric function random variable is the mutual information of the corresponding channel, while its variance equals the varentropy of the channel -- two parameters that are particularly relevant in finite block-length regimes. Additionally, we show that after polarization, the variances of the bit metrics approach zero for binary-input discrete memoryless channels (BI-DMCs). Moreover, we provide an estimate for calculating the variance of the binary-input additive white Gaussian noise (BI-AWGN) channel. We introduce a list-pruning strategy for FSCL decoding that retains only the paths whose metric values are close to the average. As a result, our method significantly reduces the number of required sorting operations in FSCL-based decoding algorithms. We also derive an upper bound, as a function of the polarized channel varentropy, on the probability that the distance between a bit-metric random variable and the bit-channel mutual information exceeds a given threshold. Leveraging this result, we further propose a varentropy-based list-pruning strategy for the SCL (VPSCL) decoding algorithm that adapts to the varentropy of the corresponding bit-channel. Our proposed pruning strategy also benefits stack decoding (VPStack) by discarding partial paths and avoiding unnecessary extensions.