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A Method to Reduce the Complexity of Computing the Complete Weight Distribution of Polar Codes

Zhichao Liu, Zhiming Ma, Guiying Yan

TL;DR

This work defines a new subgroup of LTA of decreasing monomial codes which can find more cosets with the same weight distribution and further reduces the complexity by proofing the group action on a coset set is transitive.

Abstract

The code spectrum of polar codes is crucial to the performance of polar codes. Based on the lower-triangular affine group (LTA) of decreasing monomial codes and the one-variable descendance (ovd) relation, we define a new subgroup of LTA which can find more cosets with the same weight distribution. Using this algebraic structure, we further reduce the complexity by proofing the group action on a coset set is transitive. Our method is an enhanced version of previous research, and the complexity of most cases can be reduced exceeding several times.

A Method to Reduce the Complexity of Computing the Complete Weight Distribution of Polar Codes

TL;DR

This work defines a new subgroup of LTA of decreasing monomial codes which can find more cosets with the same weight distribution and further reduces the complexity by proofing the group action on a coset set is transitive.

Abstract

The code spectrum of polar codes is crucial to the performance of polar codes. Based on the lower-triangular affine group (LTA) of decreasing monomial codes and the one-variable descendance (ovd) relation, we define a new subgroup of LTA which can find more cosets with the same weight distribution. Using this algebraic structure, we further reduce the complexity by proofing the group action on a coset set is transitive. Our method is an enhanced version of previous research, and the complexity of most cases can be reduced exceeding several times.

Paper Structure

This paper contains 20 sections, 11 theorems, 56 equations, 3 tables.

Table of Contents

  1. Introduction
  2. PRELIMINARIES
  3. Polar codes and polar cosets
  4. Decreasing monomial codes
  5. LTA and its subgroup
  6. Reduce complexity using the group action
  7. An enhanced version in reducing complexity by a new subgroup of LTA
  8. A new subgroup of LTA
  9. The main theorem
  10. An example
  11. Applications of our enhanced method
  12. Conclusion
  13. the group $LTA(n,2)_{h_2}^{h_1}$ actions on $h_1$
  14. the group $LTA(n,2)_{h_2}^{h_1}$ actions on $h_2$
  15. the group $LTA(n,2)_{h_2}^{h_1}$ actions on $\mathcal{O}_{h_2}\backslash \mathcal{O}_{h_1}\bigcap \mathcal{D}$
  16. ...and 5 more sections

Key Result

Proposition 2.1

$\mathcal{P}(N,K)$ can be denoted by

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 14 more