A design methodology is proposed to optimize the minimum weight and error coefficient of polarization-adjusted convolutional (PAC) codes, enhancing their maximum likelihood (ML) performance based on a theoretical analysis of code asymptotic behavior. Employing an adapted multilevel list search algorithm to identify minimum-weight codewords, the methodology comprises three deterministic optimization algorithms. First, an iterative rate-profiling optimization algorithm substantially reduces the number of minimum-weight codewords through efficient pairwise exchanges of information and frozen indices. Second, a tree search optimization algorithm progressively extends the convolutional impulse response, exploring superior solutions within a theoretically constrained search space. Third, a joint optimization algorithm synthesizes the two algorithms, alternately refining the rate-profiling and convolutional pre-transform. Complexity analysis underscores the computational efficiency of these algorithms for short PAC codes, while optimization results confirm the strong capability of the proposed methodology in improving the minimum weight and error coefficient. With moderate-to-large list decoding for code lengths of 64 to 256, the proposed PAC codes consistently outperform state-of-the-art polar code variants, attaining or approaching the random coding union (RCU) bound. Additionally, the proposed PAC codes demonstrate the capability to exceed the normal approximation (NA) bound at low-to-moderate code rates.

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.