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

2026-02-02

AI Summary

This work addresses the problem of efficient near-maximum-likelihood decoding for Golay codes and their associated lattices, such as the Leech lattice Λ₂₄ and its main sublattice H₂₄. The authors propose a parallel list decoding algorithm that integrates the conceptual framework of polarization-adjusted convolutional (PAC) codes with Forney’s cubic construction. This approach marks the first application of the PAC coding paradigm to Golay code decoding, eliminating the need for index permutations and codeword puncturing required in conventional methods. Consequently, the decoding process is significantly simplified while achieving enhanced parallelism. The proposed algorithm maintains performance close to maximum likelihood and naturally extends to high-dimensional lattice structures, thereby enabling a unified and efficient decoding strategy for both Golay codes and the Leech lattice.

Technology Category

• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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.

Problem

• Golay codes
• lattice decoding
• Leech lattice
• PAC codes
• maximum likelihood decoding

Innovation

• PAC codes
• Golay codes
• parallel list decoding
• Leech lattice
• Forney's cubing construction