[Paper Review] On the Closed-form Weight Enumeration of Polar Codes: 1.5$d$-weight Codewords

Mohammad Rowshan, Vlad Drăgoi+1 more|arXiv (Cornell University)|May 4, 2023

Coding theory and cryptography5 citations

TL;DR

This paper presents a closed-form enumeration formula for 1.5×w_min-weight codewords in polar codes and other decreasing monomial codes, leveraging the lower triangular affine (LTA) group and Minkowski sums of orbits. By analyzing intersections of kernel sets and applying a collision-discounting mechanism, the authors derive an exact count that depends only on the maximum-degree monomials, extending prior work limited to minimum-weight codewords.

ABSTRACT

The weight distribution of error correction codes is a critical determinant of their error-correcting performance, making enumeration of utmost importance. In the case of polar codes, the minimum weight $\wm$ (which is equal to minimum distance $d$) is the only weight for which an explicit enumerator formula is currently available. Having closed-form weight enumerators for polar codewords with weights greater than the minimum weight not only simplifies the enumeration process but also provides valuable insights towards constructing better polar-like codes. In this paper, we contribute towards understanding the algebraic structure underlying higher weights by analyzing Minkowski sums of orbits. Our approach builds upon the lower triangular affine (LTA) group of decreasing monomial codes. Specifically, we propose a closed-form expression for the enumeration of codewords with weight $1.5\wm$. Our simulations demonstrate the potential for extending this method to higher weights.

Motivation & Objective

• To address the long-standing challenge of enumerating higher-weight codewords in polar codes beyond the minimum weight.
• To extend the known closed-form formula for minimum-weight codewords to codewords of weight 1.5×w_min.
• To develop a systematic framework for weight enumeration in decreasing monomial codes using algebraic structures and group actions.
• To provide insights into the algebraic properties of polar codes that can inform better code construction and decoding.

Proposed method

• The method builds on the lower triangular affine (LTA) group of decreasing monomial codes to analyze the structure of codewords.
• It uses Minkowski sums of orbits to model the formation of 1.5×w_min-weight codewords from pairs of minimum-weight codewords.
• A classification theorem by Kasami and Tokura is applied to characterize codewords with weight less than 2×w_min.
• The number of such codewords is computed via a formula that accounts for kernel set intersections and avoids overcounting due to collisions.
• The key equation is A_{1.5w_min}{f,g} = 2^{|K_f| + |K_g| - (r-2) - |K_f ∩ K_g|}, which corrects for overlapping rows in cosets.
• The approach relies on identifying matching binary digits in the support of codewords from different cosets to ensure correct weight formation.

Experimental results

Research questions

• RQ1 How can codewords of weight 1.5×w_min be systematically enumerated in polar codes using algebraic structures?
• RQ2 What role does the LTA group play in the formation and enumeration of higher-weight codewords?
• RQ3 How do kernel set intersections between cosets lead to overcounting, and how can this be corrected in the enumeration process?
• RQ4 Can the closed-form formula for minimum-weight codewords be extended to codewords of weight 1.5×w_min?
• RQ5 What is the dependence of the 1.5×w_min-weight codeword count on the maximum-degree monomials in the code's generator structure?

Key findings

• The paper derives a closed-form expression for the number of 1.5×w_min-weight codewords in decreasing monomial codes, valid for any polar code.
• The number of such codewords is given by A_{1.5w_min}{f,g} = 2^{|K_f| + |K_g| - (r-2) - |K_f ∩ K_g|}, which accounts for kernel set sizes and their intersection.
• For the (128,64) polar code with m=7, r=4, and f=104, g=84, the number of 1.5×w_min-weight codewords is 512.
• When |ind(g)\ind(f)| = 2 and r=2, the formula simplifies to A_{1.5w_min}{f,g} = 2^4 × 2^4 × 2^{-2} = 64.
• The method correctly handles collisions due to overlapping rows in kernel sets, reducing overcounting by discounting 2^{|K_f ∩ K_g|} redundant codewords.
• The framework demonstrates potential for extension to higher weights, such as 2×w_min and beyond, via recursive application of the same principles.