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

TL;DR

This paper introduces a novel decoding approach for Golay codes using PAC codes, enabling efficient decoding without index permutation or puncturing, and extends this method to related lattices like the Leech lattice.

Contribution

It presents a new decoding algorithm for Golay codes based on PAC codes, improving efficiency and simplifying the decoding process compared to prior methods.

Findings

Achieved near-maximum likelihood decoding performance.

Eliminated the need for index permutation and codeword puncturing.

Extended decoding techniques to related lattices such as the Leech lattice.

Abstract

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 $\Lambda_{24}$ and its principal sublattice $H_{24}$, can be also decoded efficiently.