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On the Weight Distribution of Weights Less than $2w_{\min}$ in Polar Codes

Zicheng Ye, Yuan Li, Huazi Zhang, Jun Wang, Guiying Yan, Zhiming Ma

TL;DR

The paper tackles the problem of precisely characterizing the number of low-weight codewords in decreasing polar codes, a task critical for accurate ML-based decoding bounds. It extends Kasami–Tokura style results from RM codes to decreasing polar codes by developing a unified enumeration framework based on restricted polynomial forms and largest-term classifications, enabling closed-form counts for codewords with weights below $2w_{\min}$ and guaranteeing polynomial-time complexity in code length $N=2^m$. The main technical contributions are the restricted-form decomposition (yielding unique factorizations), and the detailed Type-I and Type-II counting formulas with corresponding subfunctions and complexity analyses, culminating in a complete formula for $M=A_{2^{m-r+1}-2^{m-r+1-\mu}}(C(\mathcal{I}))$ for $\mu\ge1$. The findings offer theoretical insight into the weight spectrum of polar codes and practical tools for evaluating union bounds and decoding performance, with numerical validation against RM codes and example polar constructions, and they demonstrate polynomial-time feasibility even for large $m$ and $r$.

Abstract

The number of low-weight codewords is critical to the performance of error-correcting codes. In 1970, Kasami and Tokura characterized the codewords of Reed-Muller (RM) codes whose weights are less than $2w_{\min}$, where $w_{\min}$ represents the minimum weight. In this paper, we extend their results to decreasing polar codes. We present the closed-form expressions for the number of codewords in decreasing polar codes with weights less than $2w_{\min}$. Moreover, the proposed enumeration algorithm runs in polynomial time with respect to the code length.

On the Weight Distribution of Weights Less than $2w_{\min}$ in Polar Codes

TL;DR

The paper tackles the problem of precisely characterizing the number of low-weight codewords in decreasing polar codes, a task critical for accurate ML-based decoding bounds. It extends Kasami–Tokura style results from RM codes to decreasing polar codes by developing a unified enumeration framework based on restricted polynomial forms and largest-term classifications, enabling closed-form counts for codewords with weights below and guaranteeing polynomial-time complexity in code length . The main technical contributions are the restricted-form decomposition (yielding unique factorizations), and the detailed Type-I and Type-II counting formulas with corresponding subfunctions and complexity analyses, culminating in a complete formula for for . The findings offer theoretical insight into the weight spectrum of polar codes and practical tools for evaluating union bounds and decoding performance, with numerical validation against RM codes and example polar constructions, and they demonstrate polynomial-time feasibility even for large and .

Abstract

The number of low-weight codewords is critical to the performance of error-correcting codes. In 1970, Kasami and Tokura characterized the codewords of Reed-Muller (RM) codes whose weights are less than , where represents the minimum weight. In this paper, we extend their results to decreasing polar codes. We present the closed-form expressions for the number of codewords in decreasing polar codes with weights less than . Moreover, the proposed enumeration algorithm runs in polynomial time with respect to the code length.
Paper Structure (13 sections, 14 theorems, 40 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 13 sections, 14 theorems, 40 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polar codes as monomial codes
  4. Decreasing monomial codes
  5. Weight distribution of RM codes
  6. Weight distribution of polar codes
  7. Proof Outline
  8. Classification and enumeration of polynomials
  9. Restricted form of polynomials
  10. Number of the type-I polynomials
  11. Number of the type-II polynomials
  12. Number of Low-weight polynomials
  13. Conclusion

Key Result

Theorem 1

Let $f$ be a non-zero polynomial in $\operatorname{RM}(m, r)$. If the weight of $f$ is less than $2^{m-r+1}$, then $\mathrm{wt}(f) = 2^{m-r+1}-2^{m-r+1-\mu}$, where $\mu$ is a positive integer. Moreover, $f$ can be written as one of the following forms: or Herehere $g_1,g_2,\cdots$ are linearly independent linear polynomials. We call the Equation (eq1) the type-I polynomials, and Equation (eq2)

Figures (3)

  • Figure 1: SCL performance and union bound for different [128, 80] polar codes
  • Figure : Calculate $|A_{\mu}|$
  • Figure : Calculate $|B_{\mu}|$

Theorems & Definitions (44)

  • Example 1
  • Definition 1
  • Example 2
  • Definition 2
  • Theorem 1: Kasami1970
  • Example 3
  • Example 4
  • Lemma 1
  • proof
  • Example 5
  • ...and 34 more