Reverse PAC Codes: Look-ahead List Decoding
Xinyi Gu, Mohammad Rowshan, Jinhong Yuan
TL;DR
This work targets the reduction of minimum-weight codewords $A_{w_{\min}}$ in polar-type codes to improve block error rate (BLER) at high rates. It introduces reverse PAC (RPAC) codes that apply a reverse pre-transform $u=\mathbf{v}P'$ (with $P'=P^{T}$) to shift coset leaders and further reduce $A_{w_{\min}}$, and develops a look-ahead list decoding (LA-SCL) algorithm to decode RPAC without increasing overall complexity to $O(L N \log_2 N)$. The paper provides a practical mapping for the pre-transform, analyzes coset structure, and proves that RPAC can significantly reduce $A_{w_{\min}}$ compared to PAC and polar codes, while LA-SCL maintains tractable decoding. Numerical results for high-rate short codes demonstrate up to ~0.6 dB gains over PAC/CRC-polar at high SNR, with RPAC approaching near-ML performance as the list size grows, illustrating the practical impact for future high-rate communications.
Abstract
Convolutional precoding in polarization-adjusted convolutional (PAC) codes is a recently introduced variant of polar codes. It has demonstrated an effective reduction in the number of minimum weight codewords (a.k.a error coefficient) of polar codes. This reduction has the potential to significantly improve the error correction performance. From a codeword formation perspective, this reduction has limitations. Capitalizing on the understanding of the decomposition of minimum-weight codewords, this paper studies reverse precoding that can effectively reduce minimum-weight codewords more than in PAC codes. We propose a look-ahead list decoding for the reverse PAC codes, which has the same order of complexity as list decoding in PAC codes. Through numerical analysis, we demonstrate a notable reduction in error coefficients compared to PAC codes and polar codes, resulting in a remarkable improvement in the block error rate, in particular at high code rates.