Error Coefficient-improved Polar/PAC Codes

Polar codes are normally designed based on the reliability of the sub-channels in the polarized vector channel. There are various methods with diverse complexity and accuracy to evaluate the reliability of the sub-channels. However, designing polar codes solely based on the sub-channel reliability may result in poor Hamming distance properties. In this work, we propose a different approach to design the information set for polar codes and PAC codes where the objective is to reduce the number of codewords with minimum weight (a.k.a. error coefficient) of a code designed for maximum reliability. This approach is based on the coset-wise characterization of the rows of polar transform $\mathbf{G}_N$ involved in the formation of the minimum-weight codewords. Our analysis capitalizes on the properties of the polar transform based on its row and column indices. The numerical results show that the designed codes outperform PAC codes and CRC-Polar codes at the practical block error rate of $10^{-2}-10^{-3}$. Furthermore, a by-product of the combinatorial properties analyzed in this paper is an alternative enumeration method of the minimum-weight codewords.