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On the Weight Spectrum of Rate-Compatible Polar Codes

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

TL;DR

This work addresses the challenging problem of the weight spectrum for rate-compatible polar codes by developing a unified algebraic framework that treats polar codes as decreasing monomial codes and leverages automorphisms and pre-transformations. It provides exact enumeration of minimum-weight codewords for bit-reversal shortened, QUP, and Wang-Liu shortened decreasing polar codes, and introduces polynomial-time algorithms to compute the average weight spectrum under random upper-triangular pre-transformations. The key contributions include explicit formulas for minimum-weight counts, recursive and efficient methods for punctured/shortened spectra, and an ensemble-based approach to average spectra with complexity $O(N^3)$. Numerical results validate that the proposed methods yield accurate performance estimates and tight union bounds, enabling scalable analysis of rate-compatible polar codes for practical block lengths.

Abstract

The weight spectrum plays a crucial role in the performance of error-correcting codes. Despite substantial theoretical exploration of polar codes with mother code length, a framework for the weight spectrum of rate-compatible polar codes remains elusive. In this paper, we address this gap by presenting the theoretical results for enumerating the number of minimum-weight codewords for quasi-uniform punctured, Wang-Liu shortened, and bit-reversal shortened decreasing polar codes. Additionally, we propose efficient algorithms for computing the average spectrum of random upper-triangular pre-transformed shortened and punctured polar codes. Notably, our algorithms operate with polynomial complexity relative to the code length. Simulation results affirm that our findings yield a precise estimation of the performance of rate-compatible polar codes.

On the Weight Spectrum of Rate-Compatible Polar Codes

TL;DR

This work addresses the challenging problem of the weight spectrum for rate-compatible polar codes by developing a unified algebraic framework that treats polar codes as decreasing monomial codes and leverages automorphisms and pre-transformations. It provides exact enumeration of minimum-weight codewords for bit-reversal shortened, QUP, and Wang-Liu shortened decreasing polar codes, and introduces polynomial-time algorithms to compute the average weight spectrum under random upper-triangular pre-transformations. The key contributions include explicit formulas for minimum-weight counts, recursive and efficient methods for punctured/shortened spectra, and an ensemble-based approach to average spectra with complexity . Numerical results validate that the proposed methods yield accurate performance estimates and tight union bounds, enabling scalable analysis of rate-compatible polar codes for practical block lengths.

Abstract

The weight spectrum plays a crucial role in the performance of error-correcting codes. Despite substantial theoretical exploration of polar codes with mother code length, a framework for the weight spectrum of rate-compatible polar codes remains elusive. In this paper, we address this gap by presenting the theoretical results for enumerating the number of minimum-weight codewords for quasi-uniform punctured, Wang-Liu shortened, and bit-reversal shortened decreasing polar codes. Additionally, we propose efficient algorithms for computing the average spectrum of random upper-triangular pre-transformed shortened and punctured polar codes. Notably, our algorithms operate with polynomial complexity relative to the code length. Simulation results affirm that our findings yield a precise estimation of the performance of rate-compatible polar codes.

Paper Structure

This paper contains 18 sections, 13 theorems, 38 equations, 10 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polar codes as monomial codes
  4. Decreasing monomial codes
  5. Pre-Transformed Polar Codes
  6. minimum-weight codewords of decreasing monomial codes
  7. Automorphism group of decreasing monomial codes
  8. Puncturing and shortening
  9. minimum-weight codewords of punctured and shortened polar codes
  10. Notations and Definitions
  11. Number of minimum-weight codewords in the bit-reversal shortened polar codes
  12. Number of minimum-weight codewords in QUP and Wang-Liu shortened polar codes
  13. Weight spectrum of $T_P(f,X_i)$
  14. Number of minimum-weight codewords in QUP polar codes
  15. Number of minimum-weight codewords in Wang-Liu shortened polar codes
  16. ...and 3 more sections

Key Result

Theorem 1

Bardet2016 Let $C(\mathcal{I})$ be an $r$-th order decreasing monomial code. Then $T_{\mathcal{F}}(x_{i_1}\cdots x_{i_r})$ consists of the codewords where $B(f,j)$ is the set $\{1\leq k\leq m \mid k<i_j, k\neq i_1,\dots, i_r\}$. Define $\lambda_{x_{i_1}\cdots x_{i_r}} = 2^{\sum_{t=1}^r (i_t-t+1)}$, then the number of codewords in $T_{\mathcal{F}}(x_{i_1}\cdots x_{i_r})$ is $\lambda_{x_{i_1}\cdots

Figures (10)

  • Figure 1: SCL performance and union bound for different [96,24] polar codes
  • Figure 2: SCL performance and union bound for different [96,48] polar codes
  • Figure 3: SCL performance and union bound for different [96,72] polar codes
  • Figure 4: SCL performance and union bound for different [768,192] polar codes
  • Figure 5: SCL performance and union bound for different [768,384] polar codes
  • ...and 5 more figures

Theorems & Definitions (38)

  • Definition 1
  • Example 1
  • Theorem 1
  • Example 2
  • Definition 2: Punctured Code and Puncturing Pattern
  • Definition 3: Shortened Code and Shortening Pattern
  • Example 3
  • Remark 1
  • Example 4
  • Lemma 1
  • ...and 28 more