Decoding Golay Codes and their Related Lattices: A PAC Code Perspective

In this work, we propose a decoding method of Golay codes from the perspective of Polarization Adjusted Convolutional (PAC) codes. By invoking Forney’s cubing construction of Golay codes and their generators $G^*(8,7)/(8,4)$, we found different construction methods of Golay codes from PAC codes, which result in an efficient parallel list decoding algorithm with near-maximum likelihood performance. Compared with existing methods, our method can get rid of index permutation and codeword puncturing. Using the new decoding method, some related lattices, such as Leech lattice $Λ_{24}$ and its principal sublattice $H_{24}$, can be also decoded efficiently.

💡 Research Summary

The paper introduces a novel decoding framework for the binary Golay codes and related lattices (the Leech lattice Λ₍₂₄₎ and its principal sublattice H₍₂₄₎) by exploiting the structure of Polarization‑Adjusted Convolutional (PAC) codes. Starting from Forney’s cubing construction, the authors express the (24, 12, 8) extended Golay code using the generator matrices G* (8, 7) and G* (8, 4). They show that the Golay generator can be written as a product of a standard polar generator matrix Gₚ and an upper‑triangular convolutional matrix T, thus embedding the Golay code into a PAC‑code architecture without any index permutation or puncturing.

Three distinct 3 × 3 kernels are considered:

1. F(1)₃ =