On Convolutional Precoding in PAC Codes

TL;DR

This paper analyzes how convolutional precoding in PAC codes reduces minimum-weight codewords, discusses its limitations, and proposes irregular precoding methods to enhance error correction performance.

Contribution

It explicitly explains the impact of convolutional precoding on codeword weight distribution and introduces irregular precoding strategies for improvement.

Findings

Convolutional precoding reduces the number of minimum-weight codewords.

Precoding effectiveness varies depending on the code segment.

Irregular precoding can potentially improve error correction performance.

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.