Optimized Rate-Profiling for PAC Codes

The polarization-adjusted convolutional (PAC) codes concatenate the polar transform and the convolutional transform to improve the decoding performance of the finite-length polar codes, where the rate-profile is used to construct the PAC codes by setting the positions of frozen bits. However, the optimal rateprofile method of PAC codes is still unknown. In this paper, an optimized rate-profile algorithm of PAC codes is proposed. First, we propose the normalized compression factor (NCF) to quantify the transmission efficiency of useful information, showing that the distribution of useful information that needs to be transmitted after the convolutional transform should be adaptive to the capacity profile after finite-length polar transform. This phenomenon indicates that the PAC code improves the transmission efficiency of useful information, which leads to a better decoding performance than the polar codes with the same length. Then, we propose a novel rate-profile method of PAC codes, where a quadratic optimization model is established and the Euclidean norm of the NCF spectrum is adopted to construct the objective function. Finally, a heuristic bit-swapping strategy is designed to search for the frozen set with high objective function values, where the search space is limited by considering the only bits with medium Hamming weight of the row index. Simulation results show that the PAC codes with the proposed optimized rate-profile construction have better decoding performance than the PAC codes with the originally proposed Reed-Muller design construction.