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

Yujun Ji, Ling Liu, Shanxiang Lyu, Chao Chen, Tao Dai, Baoming Bai

TL;DR

The paper addresses decoding Golay codes by recasting them through PAC codes, leveraging Forney's cubing construction to build equivalent Golay generators and enabling a parallel list-decoding framework that achieves near-ML performance with small list sizes. It presents three distinct 3×3 kernel-based PAC constructions, each producing a Golay realization without index permutation or puncturing, and combines them into a unified parallel decoding scheme that selects the best result. Beyond Golay codes, the approach extends to multilevel decoding of the Leech lattice’s binary relatives, specifically $\\\\\\\\\\\\\\\\\\\\\\Lambda_{24}$ and $H_{24}$, via a lattice-structured, level-wise decoding pipeline. The results show near-ML performance with modest complexity and demonstrate practical pathways for efficient decoding of related lattice codes with potential applications in higher-dimensional modulation and post-quantum contexts.

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

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

TL;DR

The paper addresses decoding Golay codes by recasting them through PAC codes, leveraging Forney's cubing construction to build equivalent Golay generators and enabling a parallel list-decoding framework that achieves near-ML performance with small list sizes. It presents three distinct 3×3 kernel-based PAC constructions, each producing a Golay realization without index permutation or puncturing, and combines them into a unified parallel decoding scheme that selects the best result. Beyond Golay codes, the approach extends to multilevel decoding of the Leech lattice’s binary relatives, specifically and , via a lattice-structured, level-wise decoding pipeline. The results show near-ML performance with modest complexity and demonstrate practical pathways for efficient decoding of related lattice codes with potential applications in higher-dimensional modulation and post-quantum contexts.

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 , 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 and its principal sublattice , can be also decoded efficiently.
Paper Structure (17 sections, 2 theorems, 37 equations, 7 figures, 4 algorithms)

This paper contains 17 sections, 2 theorems, 37 equations, 7 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries of Polar Codes and PAC Codes
  3. Polar Codes
  4. PAC Codes
  5. Constructing Golay Codes from PAC Codes
  6. Motivation
  7. Three Methods based on Different Kernels
  8. $3\times3$ Kernel 1
  9. $3\times3$ Kernel 2
  10. $3\times3$ Kernel 3
  11. A Combined Parallel Decoding Method
  12. Simulation Results and Comparison
  13. Multilevel Decoding of $\Lambda_{24}$ and $H_{24}$
  14. Multilevel Decoding of Binary Lattices
  15. Decoding $H_{24}$
  16. ...and 2 more sections

Key Result

Proposition 1

For the generator matrix $G_8^{'(3)}$ derived from the permutation $\pi_3 = [5\ 4\ 2\ 3\ 1\ 6\ 7\ 8]$, there exists a corresponding PAC code. Its relevant parameters are given by and

Figures (7)

  • Figure 1: Parallel decoding framework of Golay codes.
  • Figure 2: BLER performance of the (24, 12, 8) extended Golay code under SCL decoding with three kernels.
  • Figure 3: BLER performance of the (24, 12, 8) extended Golay code under SCL decoding with Kernel 1, parallel SCL decoding and ML decoding.
  • Figure 4: Framework of multilevel lattice encoding and decoding.
  • Figure 5: A generator matrix of $R\Lambda_{24}$.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • proof