On Convolutional Precoding in PAC Codes

Abstract: Polarization-adjusted convolutional (PAC) codes are special concatenated codes in which we employ a one-to-one convolutional transform as a precoding step before the polar transform. In this scheme, the polar transform (as a mapper) and the successive cancellation process (as a demapper) present a synthetic vector channel to the convolutional transformation. The numerical results in the literature show that this concatenation improves the weight distribution of polar codes which justifies the superior error correction performance of PAC codes relative to polar codes. In this work, we explicitly show why the convolutional precoding reduces the number of minimumweight codewords. Further analysis exhibits where the precoding stage is not effective. Then, we recognize weaknesses of the convolutional precoding which are unequal error protection (UEP) of the information bits due to rate profiling and lack of cross-segmental convolution. Finally, we assess the possibility of improving the precoding stage by proposing some irregular convolutional precodings.