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Weight Structure of Low/High-Rate Polar Codes and Its Applications

Mohammad Rowshan, Vlad-Florin Drăgoi, Jinhong Yuan

Abstract

The structure of a linear block code is pivotal in defining fundamental properties, particularly weight distribution, and code design. In this study, we characterize the Type II structure of polar codewords with weights less than twice the minimum weight $w_{min}$, utilizing the lower triangular affine (LTA) transform. We present a closed-form formula for their enumeration. Leveraging this structure and additionally characterizing the structure of weight $2w_{min}$, we ascertain the complete weight distribution of low-rate and, through the utilization of dual codes properties, high-rate polar codes, subcodes of Reed--Muller (RM) codes, and RMxPolar codes. Furthermore, we introduce a partial order based on the weight distribution and briefly explore its properties and applications in code construction and analysis.

Weight Structure of Low/High-Rate Polar Codes and Its Applications

Abstract

The structure of a linear block code is pivotal in defining fundamental properties, particularly weight distribution, and code design. In this study, we characterize the Type II structure of polar codewords with weights less than twice the minimum weight , utilizing the lower triangular affine (LTA) transform. We present a closed-form formula for their enumeration. Leveraging this structure and additionally characterizing the structure of weight , we ascertain the complete weight distribution of low-rate and, through the utilization of dual codes properties, high-rate polar codes, subcodes of Reed--Muller (RM) codes, and RMxPolar codes. Furthermore, we introduce a partial order based on the weight distribution and briefly explore its properties and applications in code construction and analysis.
Paper Structure (22 sections, 18 theorems, 45 equations, 3 figures, 5 tables)

This paper contains 22 sections, 18 theorems, 45 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Polar codes as Decreasing monomial codes
  3. Coding theory notations
  4. Multivariate monomial and polynomials
  5. Monomial codes
  6. Decreasing monomial codes
  7. Permutation group
  8. Minimum Weight Codewords
  9. Structure of weights less than $2\mathop{\mathrm{w}}\nolimits_{\min}$
  10. Classification of Type II codewords
  11. Enumeration formulae for Type II codewords
  12. Complete weight distribution for subcodes of the second order Reed-Muller code and their duals
  13. Numerical results
  14. A partial order based on weight contribution
  15. Symmetric property and non-comparable elements
  16. ...and 7 more sections

Key Result

Theorem 1

Let $r<m$ and $P\in{\mathbf {R}}_m$ be such that $\deg(P)\leq r$ with $0<\mathop{\mathrm{w}}\nolimits(\operatorname{ev}(P))<2^{m+1-r}.$ Then $P$ is affine equivalent to one of the forms In both cases $y_i$ are linear independent forms and $\mathop{\mathrm{w}}\nolimits(\operatorname{ev}(P))=2^{m+1-r}-2^{m+1-r-\mu}.$

Figures (3)

  • Figure 1: Non-zero codewords creation for $m=6.$ Monomials in $\mathcal{I}$ (bold brown) are defined by $f\preceq x_1x_4,f\preceq x_0x_5$ and $f\preceq x_2x_4.$ For $\mathop{\mathrm{w}}\nolimits_{\min}$ each level gives $|\lambda_f|$, for $1.5\mathop{\mathrm{w}}\nolimits_{\min}$ each red arrow indicate $(f,g)$ with $\gcd(f,g)=1$, whereas for $1.75\mathop{\mathrm{w}}\nolimits_{\min}$ blue arrows indicate triplets $(f,g,h)$ with $\gcd(f,g)=\gcd(g,h)=1.$
  • Figure 2: A diagram illustrating the sequence of $|\lambda_f|$ for all $f\in \mathcal{I}_2$ and $m=7.$ The maximum value belongs to $x_5x_6$, which gives $|\lambda_{x_5x_6}|=2(m-2)=10.$ The sequence of $S_j$ is symmetric: $1,1,2,2,3,3,3,2,2,1,1.$
  • Figure 3: Five codes of dimension 19 for $m=7$. Delimited areas can be compared to different metro/bus lines, where each line has exactly 18 stops.

Theorems & Definitions (36)

  • Example 1
  • Definition 1: Monomial code
  • Definition 2
  • Definition 3
  • Definition 4: bardetdragoi17thesis
  • Definition 5
  • Example 2
  • Theorem 1: sloane1970weight,kasami1970weight
  • Theorem 2
  • Theorem 3: rowshan1.5w
  • ...and 26 more