Polarization-adjusted Convolutional (PAC) Codes: Fano Decoding vs List Decoding

In the Shannon lecture at the 2019 International Symposium on Information Theory (ISIT), Ar{\i}kan proposed to employ a one-to-one convolutional transform as a pre-coding step before polar transform. The resulting codes of this concatenation are called {\em polarization-adjusted convolutional (PAC) codes}. In this scheme, a pair of polar mapper and demapper as pre- and post-processing devices are deployed around a memoryless channel, which provides polarized information to an outer decoder leading to improved error correction performance of outer code. In this paper, the implementations of list decoding and Fano decoding for PAC codes are first investigated. Then, in order to reduce the complexity of sequential decoding of PAC/polar codes, we propose (i) an adaptive heuristic metric, (ii) tree search constraints for backtracking to avoid exploration of unlikely sub-paths, and (iii) tree search strategies consistent with the pattern of error occurrence in polar codes. These contribute to reducing the average decoding time complexity up to 85\%, with only a relatively small degradation in error correction performance. Additionally, as an important ingredient in Fano decoding of PAC/polar codes, an efficient computation method for the intermediate LLRs and partial sums is provided. This method is necessary for backtracking and avoids storing the intermediate information or restarting the decoding process.