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Toward Universal Decoding of Binary Linear Block Codes via Enhanced Polar Transformations

Chien-Ying Lin, Yu-Chih Huang, Shin-Lin Shieh, Po-Ning Chen

TL;DR

This work tackles universal soft decoding for binary linear block codes (BLBCs) by transforming any code into a polar-like structure that can be decoded with mature polar decoders. The method, PD+, combines pruning of Arıkan’s polar kernels, shortening to match target lengths, and a simulated-annealing search over the permutation, pruning, and shortening parameters to produce a decodable polar-like code. Across diverse codes (e.g., extended BCH, extended Golay, binary quadratic residue codes) PD+ with SCL decoding achieves performance that is competitive with or better than OSD, GRAND, and the original PD, while significantly reducing decoding complexity. The approach is forward-compatible with future polar-decoding advances and AI-driven search methods, offering a robust, universal BLBC decoder for current and future systems.

Abstract

Binary linear block codes (BLBCs) are essential to modern communication, but their diverse structures often require tailor-made decoders, increasing complexity. This work introduces enhanced polar decoding ($\mathsf{PD}^+$), a universal soft decoding algorithm that transforms any BLBC into a polar-like code compatible with efficient polar code decoders such as successive cancellation list (SCL) decoding. Key innovations in $\mathsf{PD}^+$ include pruning polar kernels, shortening codes, and leveraging a simulated annealing algorithm to optimize transformations. These enable $\mathsf{PD}^+$ to achieve competitive or superior performance to state-of-the-art algorithms like OSD and GRAND across various codes, including extended BCH, extended Golay, and binary quadratic residue codes, with significantly lower complexity. Moreover, $\mathsf{PD}^+$ is designed to be forward-compatible with advancements in polar code decoding techniques and AI-driven search methods, making it a robust and versatile solution for universal BLBC decoding in both present and future systems.

Toward Universal Decoding of Binary Linear Block Codes via Enhanced Polar Transformations

TL;DR

This work tackles universal soft decoding for binary linear block codes (BLBCs) by transforming any code into a polar-like structure that can be decoded with mature polar decoders. The method, PD+, combines pruning of Arıkan’s polar kernels, shortening to match target lengths, and a simulated-annealing search over the permutation, pruning, and shortening parameters to produce a decodable polar-like code. Across diverse codes (e.g., extended BCH, extended Golay, binary quadratic residue codes) PD+ with SCL decoding achieves performance that is competitive with or better than OSD, GRAND, and the original PD, while significantly reducing decoding complexity. The approach is forward-compatible with future polar-decoding advances and AI-driven search methods, offering a robust, universal BLBC decoder for current and future systems.

Abstract

Binary linear block codes (BLBCs) are essential to modern communication, but their diverse structures often require tailor-made decoders, increasing complexity. This work introduces enhanced polar decoding (), a universal soft decoding algorithm that transforms any BLBC into a polar-like code compatible with efficient polar code decoders such as successive cancellation list (SCL) decoding. Key innovations in include pruning polar kernels, shortening codes, and leveraging a simulated annealing algorithm to optimize transformations. These enable to achieve competitive or superior performance to state-of-the-art algorithms like OSD and GRAND across various codes, including extended BCH, extended Golay, and binary quadratic residue codes, with significantly lower complexity. Moreover, is designed to be forward-compatible with advancements in polar code decoding techniques and AI-driven search methods, making it a robust and versatile solution for universal BLBC decoding in both present and future systems.
Paper Structure (24 sections, 3 theorems, 25 equations, 13 figures, 4 tables, 1 algorithm)

This paper contains 24 sections, 3 theorems, 25 equations, 13 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Organization
  3. Background
  4. Decoding of BLBC
  5. Conventional Polar Codes
  6. Other Polarization Kernel and Multi-Kernel Polar Codes
  7. Polar codes with Dynamic Frozen Bits
  8. Review of Polar Decoding
  9. Proposed Enhanced Polar Decoding
  10. Motivation
  11. Challenging case
  12. Multi-kernel technique to meet the code length
  13. Search algorithm
  14. Main Result and Proposed ${\mathsf{PD}^+}$
  15. Pruned Kernel
  16. ...and 9 more sections

Key Result

Theorem 1

An $(n,k)$-BLBC can be transformed into a (possibly multi-kernel) polar code with dynamic frozen bits.

Figures (13)

  • Figure 1: Arı kan's kernel.
  • Figure 2: LHS: Recursive structure of Arı kan's polarization with $n=8$. RHS: The bit-reversed version of the same structure.
  • Figure 3: A block diagram of the proposed ${\mathsf{PD}^+}$ decoder. (Upper) Encoder: All the operations in blue boxes represent virtual components and will not affect the encoding of the original BLBC. The operations inside the black box constitutes $\mathbf{M}_{\rm DF}$. (Lower) Decoder: We can choose either to adopt a decoder for the original BLBC or a decoding algorithm for polar code.
  • Figure 4: An example of a pruned polar code with $N=8$.
  • Figure 5: Example of using pruning and shortening to get kernels of size 3 from the Arı kan's kernel.
  • ...and 8 more figures

Theorems & Definitions (9)

  • Theorem 1: Proposition 1 of PD
  • Theorem 2
  • Remark 3
  • Example 4
  • Remark 5
  • Remark 6
  • Lemma 7
  • Example 8
  • Remark 9