Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Frozen Set Design for Precoded Polar Codes

Vera Miloslavskaya, Yonghui Li, Branka Vucetic

TL;DR

The paper tackles frozen-set design for precoded polar codes decoded with SCL by introducing a low-complexity framework that uses analytical performance bounds to predict near-ML complexity. It develops a tightened lower bound and a novel approximate bound to enable efficient frozen-set optimization via a genetic algorithm, while enforcing S- and B-constraint structures to prune the search space. Numerical results for length $N=512$ show superior FER performance over state-of-the-art codes across multiple list sizes, with substantially reduced design complexity. The approach provides a reproducible, bound-driven design pathway for precoded polar codes that scales to longer codes and different decoding settings, offering practical gains for reliable communications.

Abstract

This paper focuses on the frozen set design for precoded polar codes decoded by the successive cancellation list (SCL) algorithm. We propose a novel frozen set design method, whose computational complexity is low due to the use of analytical bounds and constrained frozen set structure. We derive new bounds based on the recently published complexity analysis of SCL decoding with near maximum-likelihood (ML) performance. To predict the ML performance, we employ the state-of-the-art bounds relying on the code weight distribution. The bounds and constrained frozen set structure are incorporated into the genetic algorithm to generate optimized frozen sets with low complexity. Our simulation results show that the constructed precoded polar codes of length 512 have a superior frame error rate (FER) performance compared to the state-of-the-art codes under SCL decoding with various list sizes.

Frozen Set Design for Precoded Polar Codes

TL;DR

The paper tackles frozen-set design for precoded polar codes decoded with SCL by introducing a low-complexity framework that uses analytical performance bounds to predict near-ML complexity. It develops a tightened lower bound and a novel approximate bound to enable efficient frozen-set optimization via a genetic algorithm, while enforcing S- and B-constraint structures to prune the search space. Numerical results for length show superior FER performance over state-of-the-art codes across multiple list sizes, with substantially reduced design complexity. The approach provides a reproducible, bound-driven design pathway for precoded polar codes that scales to longer codes and different decoding settings, offering practical gains for reliable communications.

Abstract

This paper focuses on the frozen set design for precoded polar codes decoded by the successive cancellation list (SCL) algorithm. We propose a novel frozen set design method, whose computational complexity is low due to the use of analytical bounds and constrained frozen set structure. We derive new bounds based on the recently published complexity analysis of SCL decoding with near maximum-likelihood (ML) performance. To predict the ML performance, we employ the state-of-the-art bounds relying on the code weight distribution. The bounds and constrained frozen set structure are incorporated into the genetic algorithm to generate optimized frozen sets with low complexity. Our simulation results show that the constructed precoded polar codes of length 512 have a superior frame error rate (FER) performance compared to the state-of-the-art codes under SCL decoding with various list sizes.
Paper Structure (19 sections, 5 theorems, 27 equations, 14 figures, 5 tables)

This paper contains 19 sections, 5 theorems, 27 equations, 14 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polar Codes
  4. Weight Distribution of Precoded Polar Codes and Their ML Performance
  5. Complexity of SCL Decoding with Near ML Performance
  6. Frozen Bit Expressions
  7. Proposed Frozen Set Design for Precoded Polar Codes
  8. Limitations of $\bar{D}^{\mathrm{low}}_{m}$ as a Decoding Complexity Measure
  9. Derivation of $\bar{D}^{\mathrm{low}}_{m}$ in Coskun2022InfTheor
  10. Proposed Tightened Lower Bound $\bar{D}^{\mathrm{tight}}_{m}$
  11. Proposed Approximate Bound $\bar{D}^{\mathrm{apx}}_{m}$ that Combines the Tightened Lower Bound and Upper Bound
  12. Frozen Set Optimization
  13. Optimization using the Genetic Algorithm GenAlgT
  14. Complexity Reduction. S-Constraint and GenAlgTS
  15. Complexity Reduction. B-Constraint and GenAlgTB
  16. ...and 4 more sections

Key Result

Lemma 1

Let sets $I,J\subseteq [2^n]$ and integer $\widetilde{n} \leq n$ satisfy the following conditions: Then where $\widetilde{m} \triangleq | I\cap [m] |$, $\overline I \triangleq [2^n]\setminus I$, and $\mathbf{0}$ is all-zero matrix/vector.

Figures (14)

  • Figure 1: $\bar{D}_m$ for the $(128,64)$ code proposed in Coskun2022InfTheor
  • Figure 2: $S_{\min}$ of the frozen sets generated by the genetic algorithms
  • Figure 3: S-constrained frozen set structure
  • Figure 4: $\Delta$ of the frozen sets generated by the genetic algorithm
  • Figure 5: B-constrained frozen set structure
  • ...and 9 more figures

Theorems & Definitions (14)

  • Example 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Lemma 4
  • ...and 4 more