  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.
