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On Weight Enumeration and Structure Characterization of Polar Codes via Group Actions

Vlad-Florin Dragoi, Mohammad Rowshan

TL;DR

This work develops a complete algebraic description of codewords with weights below $2\,\mathrm{w}_{\min}$ in decreasing monomial codes, including polar and Reed–Muller codes, by exploiting the action of the lower triangular affine group ${\rm LTA}(m,2)$. It extends prior Type II results to a full characterization of Type I codewords through Minkowski sums of ${\rm LTA}(m,2)$-orbits, introduces a restricted subgroup ${\rm LTA}(m,2)_{f}^{g}$ to manage gcd structure, and derives closed-form weight-enumeration formulas with explicit counting via orbit cardinalities. The paper also demonstrates practical applications to Reed–Muller and polar codes, presents RMxPolar examples, and proposes refinements to weight-contribution partial orders to guide code-design choices. Overall, the results provide deeper algebraic insights into the structure of polar and decreasing monomial codes and offer tools for improved weight distribution and code-design strategies.

Abstract

In this article, we provide a complete characterization of codewords in polar codes with weights less than twice the minimum distance, using the group action of the lower triangular affine (LTA) group. We derive a closed-form formula for the enumeration of such codewords. Furthermore, we introduce an enhanced partial order based on weight contributions, offering refined tools for code design. Our results extend previous work on Type II codewords to a full description of Type I codewords and offer new insights into the algebraic structure underlying decreasing monomial codes, including polar and Reed-Muller codes.

On Weight Enumeration and Structure Characterization of Polar Codes via Group Actions

TL;DR

This work develops a complete algebraic description of codewords with weights below in decreasing monomial codes, including polar and Reed–Muller codes, by exploiting the action of the lower triangular affine group . It extends prior Type II results to a full characterization of Type I codewords through Minkowski sums of -orbits, introduces a restricted subgroup to manage gcd structure, and derives closed-form weight-enumeration formulas with explicit counting via orbit cardinalities. The paper also demonstrates practical applications to Reed–Muller and polar codes, presents RMxPolar examples, and proposes refinements to weight-contribution partial orders to guide code-design choices. Overall, the results provide deeper algebraic insights into the structure of polar and decreasing monomial codes and offer tools for improved weight distribution and code-design strategies.

Abstract

In this article, we provide a complete characterization of codewords in polar codes with weights less than twice the minimum distance, using the group action of the lower triangular affine (LTA) group. We derive a closed-form formula for the enumeration of such codewords. Furthermore, we introduce an enhanced partial order based on weight contributions, offering refined tools for code design. Our results extend previous work on Type II codewords to a full description of Type I codewords and offer new insights into the algebraic structure underlying decreasing monomial codes, including polar and Reed-Muller codes.
Paper Structure (38 sections, 22 theorems, 73 equations, 1 figure, 5 tables)

This paper contains 38 sections, 22 theorems, 73 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Weight distribution of decreasing monomial codes
  3. Decreasing monomial codes -- permutation groups
  4. Weight enumeration
  5. Contributions
  6. Starting point
  7. Type I codewords - conditions on monomials
  8. Type I codewords - Minkowski sums of orbits
  9. Applications: counting formulae and weight contribution partial orders
  10. The algebraic formalism behind polar and Reed-Muller codes
  11. Basic Concepts in Coding Theory and Notations
  12. Multivariate monomials and polynomials over $\mathbb{F}_2$
  13. Decreasing Monomial Codes
  14. Permutation group of Decreasing Monomial Codes
  15. Minimum Weight Codewords
  16. ...and 23 more sections

Key Result

Lemma 1

Let $f,g\in\mathcal{M}_{m}.$ The following holds

Figures (1)

  • Figure 1: Two distinct weights in the Minkowski sum of ${\rm LTA}(m,2)\cdot x_1x_2x_3+{\rm LTA}(m,2)\cdot x_1x_4x_5$, for $m=6.$ a) $\mu=2$ or equivalently $1.5\mathop{\mathrm{w}}\nolimits_{\min}$ Type II codeword (same as vlad1.5d), and b) $\mu=3$ or $1.75\mathop{\mathrm{w}}\nolimits_{\min}$ Type I codeword. The product $x_1x_2x_3(x_1+x_0)x_4x_5$ was refactored into $(x_0+1)x_1x_2x_3x_4x_5$ which produces a vector of weight $1.$

Theorems & Definitions (48)

  • Definition 1
  • Definition 2: dragoi17thesis
  • Definition 3: bardet
  • Definition 4
  • Lemma 1
  • Definition 5: dragoi17thesisbardet
  • Definition 6: bardetdragoi17thesis
  • Definition 7: bardetdragoi17thesis
  • Theorem 1: bardetdragoi17thesis
  • Theorem 2: sloane1970weight,kasami1970weight
  • ...and 38 more