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Generalized Weight Structure of Polar Codes: Selected Template Polynomials

Mohammad Rowshan, Vlad-Florin Dragoi

Abstract

Polar codes can be viewed as decreasing monomial codes, revealing a rich algebraic structure governed by the lower-triangular affine (LTA) group. We develop a general framework to compute the Hamming weight of codewords generated by sums of monomials, express these weights in a canonical dyadic form, and derive closed expressions for key structural templates (disjoint sums, nested blocks, complementary flips) that generate the low and intermediate weight spectrum. Combining these templates with the LTA group action, we obtain explicit multiplicity formulas, yielding a unified algebraic method to characterize and enumerate codewords.

Generalized Weight Structure of Polar Codes: Selected Template Polynomials

Abstract

Polar codes can be viewed as decreasing monomial codes, revealing a rich algebraic structure governed by the lower-triangular affine (LTA) group. We develop a general framework to compute the Hamming weight of codewords generated by sums of monomials, express these weights in a canonical dyadic form, and derive closed expressions for key structural templates (disjoint sums, nested blocks, complementary flips) that generate the low and intermediate weight spectrum. Combining these templates with the LTA group action, we obtain explicit multiplicity formulas, yielding a unified algebraic method to characterize and enumerate codewords.
Paper Structure (26 sections, 11 theorems, 128 equations)

This paper contains 26 sections, 11 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Monomial Representation and Decreasing Sets
  3. Boolean functions and monomials
  4. Monomial codes and decreasing sets
  5. LTA group and orbits
  6. A General Weight Formula for Polynomials
  7. Structural Templates for Weights
  8. Nesting (weight scaling)
  9. Enumeration of Templates' Codewords
  10. General orbit-counting scheme for weight templates
  11. Head contribution.
  12. Head contribution.
  13. Kernel decomposition: $Q_{\max}$ and $R$.
  14. Tail contribution and stabilizer (collisions)
  15. Tail and linear degrees of freedom on the kernel.
  16. ...and 11 more sections

Key Result

Proposition 1

Let $P(x)=h(x)\sum_{i=1}^q f_i(x)$ be as in eq:P-factorised, with $\deg(P)=r$ and $d=2^{m-r}$. Then where

Theorems & Definitions (29)

  • Definition 1: Monomial code
  • Definition 2: Decreasing set and decreasing monomial code
  • Definition 3: Partition weights $|\lambda_f|$ and $|\lambda_f (g)|$
  • Proposition 1: General weight formula
  • Lemma 1: Dyadic decomposition
  • Lemma 2: Disjoint $k$-sum
  • proof
  • Remark 1
  • Example 1
  • Lemma 3: rank-$\ell$ and a degree drop
  • ...and 19 more