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Enumeration of Minimum Weight Codewords of Pre-Transformed Polar Codes by Tree Intersection

Andreas Zunker, Marvin Geiselhart, Stephan ten Brink

TL;DR

This work tackles the finite-length performance of pre-transformed polar codes by exactly enumerating minimum-weight codewords through a novel tree-intersection approach. The method represents the relevant message sets as PDBTs and computes their intersection to count weight-minimizing codewords with dramatically reduced complexity compared to prior works. A theoretical lower bound on the minimum distance is established, and tight results are given for RM-based PTPCs, with explicit formulas for A_{d_min}, including a closed form for RM(r,n) where r ≤ n−2. The authors apply the algorithm to long codes and optimize PAC precoders by identifying polynomials that minimize the number of minimum-weight codewords, enabling distance-aware PTPC design in practice. Overall, the paper provides a scalable, exact tool for distance-spectrum-aware PTPC design and demonstrates meaningful gains in both theory and PAC polynomial optimization.

Abstract

Pre-transformed polar codes (PTPCs) form a class of codes that perform close to the finite-length capacity bounds. The minimum distance and the number of minimum weight codewords are two decisive properties for their performance. In this work, we propose an efficient algorithm for determining the number of minimum weight codewords of general PTPCs that eliminates all redundant visits to nodes of the search tree, thus reducing the computational complexity typically by several orders of magnitude compared to state-of-the-art algorithms. This reduction in complexity allows, for the first time, the minimum distance properties to be directly considered in the code design of PTPCs. The algorithm is demonstrated for randomly pre-transformed Reed-Muller (RM) codes and polarization-adjusted convolutional (PAC) codes. Furthermore, we design optimal polynomials for PAC codes with this algorithm, minimizing the number of minimum weight codewords.

Enumeration of Minimum Weight Codewords of Pre-Transformed Polar Codes by Tree Intersection

TL;DR

This work tackles the finite-length performance of pre-transformed polar codes by exactly enumerating minimum-weight codewords through a novel tree-intersection approach. The method represents the relevant message sets as PDBTs and computes their intersection to count weight-minimizing codewords with dramatically reduced complexity compared to prior works. A theoretical lower bound on the minimum distance is established, and tight results are given for RM-based PTPCs, with explicit formulas for A_{d_min}, including a closed form for RM(r,n) where r ≤ n−2. The authors apply the algorithm to long codes and optimize PAC precoders by identifying polynomials that minimize the number of minimum-weight codewords, enabling distance-aware PTPC design in practice. Overall, the paper provides a scalable, exact tool for distance-spectrum-aware PTPC design and demonstrates meaningful gains in both theory and PAC polynomial optimization.

Abstract

Pre-transformed polar codes (PTPCs) form a class of codes that perform close to the finite-length capacity bounds. The minimum distance and the number of minimum weight codewords are two decisive properties for their performance. In this work, we propose an efficient algorithm for determining the number of minimum weight codewords of general PTPCs that eliminates all redundant visits to nodes of the search tree, thus reducing the computational complexity typically by several orders of magnitude compared to state-of-the-art algorithms. This reduction in complexity allows, for the first time, the minimum distance properties to be directly considered in the code design of PTPCs. The algorithm is demonstrated for randomly pre-transformed Reed-Muller (RM) codes and polarization-adjusted convolutional (PAC) codes. Furthermore, we design optimal polynomials for PAC codes with this algorithm, minimizing the number of minimum weight codewords.
Paper Structure (16 sections, 12 theorems, 37 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 16 sections, 12 theorems, 37 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Error Probability of Binary Linear Block Codes
  5. Polar Codes
  6. Pre-Transformed Polar Codes (PTPCs)
  7. Counting Minimum Weight Codewords of Pre-Transformed Polar Codes
  8. Minimum Distance of PTPC
  9. Minimum Weight Codewords of PTPC
  10. Message Trees
  11. Tree Intersection
  12. Theoretical Findings
  13. Results
  14. Algorithmic Complexity
  15. Number of Minimum Weight Codewords of PTPC
  16. ...and 1 more sections

Key Result

Lemma 1

A set of binary vectors $\mathcal{B} \subseteq \mathbb{F}_2^N$ can be represented by the full paths through a PDBT of height $N$ if and only if $b_0=b_{{0}}'$ for all ${\boldsymbol{b},\boldsymbol{b}' \in \mathcal{B}}$ and $\mathcal{B}$ has cardinality $\left|\mathcal{B}\right|=2^{\left|\mathcal{L}^* is the set of levels that contain sibling nodes.

Figures (4)

  • Figure 1: Block diagram of pre-transformed polar encoding.
  • Figure 2: Message trees of a universal polar coset (a) and a plain polar coset (b) for $N=16$ and $i=10$. The intersection of the trees (i.e., common messages $\boldsymbol{u}$) is highlighted. In this example, $\left|\mathcal{Q}_{i=10,\mathop{\mathrm{\textit{w}_\mathrm{min}}}\nolimits=4}(\mathcal{I})\right|=4$.
  • Figure 3: Complexity comparison of the $\mathop{\mathrm{\textit{d}_\mathrm{min}}}\nolimits$-weight codeword enumeration for $\mathop{\mathrm{\mathcal{RM}}}\nolimits((n-1)/2,n)$ rate-profile PAC codes with polynomial $p(x)\mathrel{\widehat{=}}5767471_8$. Note that the pre-transform checks coincide with the message updates in PartialEnumPAC.
  • Figure 4: Number of $d_\mathrm{min}$-weight codewords of convolutional (PAC) and randomly pre-transformed $\mathop{\mathrm{\mathcal{RM}}}\nolimits((n-1)/2,n)$ codes with $32 \le N \le 2097152$.

Theorems & Definitions (26)

  • Definition 1: DBT
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 16 more