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Algebraic Properties of PAC Codes

Vlad-Florin Dragoi, Mohammad Rowshan

TL;DR

This work extends the algebraic framework for polarization-based codes by embedding PAC and related constructions into generalized polynomial polar codes. It establishes that the dual of an upper polynomial polar code is a lower polynomial polar code and that minimum distance is preserved from the plain polar code, while offering concrete bounds on the number of minimum-weight codewords. By developing notions of monomial and polynomial subcodes, LTA-based permutations, and a backward-compatibility matrix $\mathbf{M}$, the paper provides systematic tools for characterizing duals, weight spectra, and monomial substructures in PAC-like codes. The results yield structural insights and practical guidelines for code design and analysis, including how PAC codes interact with polar structures and how to bound their distance properties. Overall, the work unifies PAC codes with a broader algebraic code family, enabling rigorous duality and weight-distribution analyses with implications for short-block-length performance and decoding strategies.

Abstract

We analyze polarization-adjusted convolutional codes using the algebraic representation of polar and Reed-Muller codes. We define a large class of codes, called generalized polynomial polar codes which include PAC codes and Reverse PAC codes. We derive structural properties of generalized polynomial polar codes, such as duality, minimum distance. We also deduce some structural limits in terms of number of minimum weight codewords, and dimension of monomial sub-code.

Algebraic Properties of PAC Codes

TL;DR

This work extends the algebraic framework for polarization-based codes by embedding PAC and related constructions into generalized polynomial polar codes. It establishes that the dual of an upper polynomial polar code is a lower polynomial polar code and that minimum distance is preserved from the plain polar code, while offering concrete bounds on the number of minimum-weight codewords. By developing notions of monomial and polynomial subcodes, LTA-based permutations, and a backward-compatibility matrix , the paper provides systematic tools for characterizing duals, weight spectra, and monomial substructures in PAC-like codes. The results yield structural insights and practical guidelines for code design and analysis, including how PAC codes interact with polar structures and how to bound their distance properties. Overall, the work unifies PAC codes with a broader algebraic code family, enabling rigorous duality and weight-distribution analyses with implications for short-block-length performance and decoding strategies.

Abstract

We analyze polarization-adjusted convolutional codes using the algebraic representation of polar and Reed-Muller codes. We define a large class of codes, called generalized polynomial polar codes which include PAC codes and Reverse PAC codes. We derive structural properties of generalized polynomial polar codes, such as duality, minimum distance. We also deduce some structural limits in terms of number of minimum weight codewords, and dimension of monomial sub-code.
Paper Structure (16 sections, 15 theorems, 33 equations, 3 figures, 5 tables)

This paper contains 16 sections, 15 theorems, 33 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Decreasing monomial codes
  3. Multivariate monomial and polynomials
  4. Monomial codes
  5. Decreasing monomial codes
  6. Permutation group
  7. Minimum Weight Codewords
  8. Generalized polynomial polar codes
  9. Definitions and Properties
  10. Upper polynomial polar codes
  11. Duality
  12. Properties of duals of monomial codes
  13. Dual of polynomial polar codes
  14. Properties of $\mathbf{M}$
  15. Dual of upper polynomial polar code
  16. ...and 1 more sections

Key Result

Proposition 1

Let $\mathcal{C}_{\mathbf{P}\boldsymbol{G}_N}(\mathcal{A})$ be a upper polynomial polar code. Then, $\mathcal{C}_{\boldsymbol{G}}([\max(\mathcal{A}^c)+1, N-1])$ is a decreasing monomial code with dimension $N-1-\max(\mathcal{A}^c)$ satisfying $\mathcal{C}_{\boldsymbol{G}}([\max(\mathcal{A}^c)+1, N-1

Figures (3)

  • Figure 1: Dimension of the intersection code between the PAC code and polar codes for $N=64$ and $14\leq k \leq 50.$
  • Figure 2: Dimension of the smallest monomial code that contains a PAC code (solid lines, above the green line) and the dimension of the largest monomial sub-code of the PAC code (dashed lines, below the green line), for $n=64$ and $14\leq k\leq 50.$
  • Figure 3: Polar code profile for dimension $k\in\{1,\dots,N\}$ (represented on rows) for $N=64.$ Red boxes are frozen indices ($\mathcal{A}^c$) while white are information indices ($\mathcal{A}$). Computed using $\beta$-expansion with $\beta=2^{1/4}.$

Theorems & Definitions (42)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: bardetdragoi17thesis
  • Definition 5
  • Remark 1
  • Example 1
  • Proposition 1
  • proof
  • Lemma 1
  • ...and 32 more