A Path Probability Perspective on Rate-Profiling Design for Polarization-Adjusted Convolutional (PAC) Codes Under List Decoding

A novel rate-profiling design is proposed for polarization-adjusted convolutional (PAC) codes under list decoding, adopting a path probability perspective. Inspired by the concepts of Hamming distance and Hamming weight, the concepts of path probability distance (PPD) and path probability weight (PPW) are innovatively introduced to measure the disparity between the erroneous paths and the correct path for PAC codes under probabilistic decoding. An in-depth analysis of the recursion of node log-likelihood ratios (LLRs) in the factor graph is conducted, elucidating the derivation of a lower bound on the PPW as a consequential outcome. Utilizing the concept of the PPW, a rate-profiling design algorithm is proposed to establish the reliability ranking for the bit-channels of PAC codes. Simulation results demonstrate that PAC codes employing the proposed rate-profiling exhibit advantages over alternative rate-profiling and other state-of-the-art polar coding schemes across various block lengths and code rates under short decoding list sizes.