#
List Decoding of Arıkan’s PAC Codes^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Brief Overview of PAC Codes

**rate-profiling**. Unlike conventional polar codes, the optimal rate-profiling choice is not known. In fact, it is not even clear what optimization criterion should govern this choice, although we hope to shed some light on this in Section 5.

#### 1.2. Our Contributions

#### 1.3. Related Work

`arxiv.org`in February 2020, while our work [50] was submitted for review in January 2020. Our results became available on

`arxiv.org`in May 2020. The Rowshan-Viterbo paper [49] was posted on

`arxiv.org`in July 2020, after our results were presented in [50].

#### 1.4. Paper Outline

## 2. Overview of Arıkan’s PAC Codes

**rate-profiling**to refer to this step, along with the choice of the index set $\mathcal{A}$.

**polar rate-profiling**, proceeds as follows. Let ${W}_{0},{W}_{1},\cdots ,{W}_{n-1}$ be the n bit-channels, defined with respect to the conventional polar code of length n. In polar rate-profiling, $\mathcal{A}$ is chosen so that $\{{W}_{i}:i\in \phantom{\rule{-0.166667em}{0ex}}\mathcal{A}\}$ consists of the k best bit-channels in terms of their capacity. In other words, the capacities of the k bit-channels $\{{W}_{i}:i\in \phantom{\rule{-0.166667em}{0ex}}\mathcal{A}\}$ are the k highest values among $I\left({W}_{0}\right),I\left({W}_{1}\right),\dots ,I\left({W}_{n-1}\right)$. The second approach proposed in [2] is called

**RM rate-profiling**. Let $wt\left(i\right)$ denote the Hamming weight of the binary expansion of an index i. In RM rate-profiling, we simply pick the k indices of the highest weight, with ties resolved arbitrarily. In other words, the set $\{wt(i):i\in \phantom{\rule{-0.166667em}{0ex}}\mathcal{A}\}$ consists of the k largest values among $wt\left(0\right),wt\left(1\right),\dots ,wt(n-1)$. Notably, without convolutional precoding, this choice of $\mathcal{A}$ generates Reed–Muller codes (as subcodes of a rate-1 polar code).

## 3. List Decoding of PAC Codes

#### 3.1. PAC Codes as Polar Codes with Dynamically Frozen Bits

Algorithm 1: List Decoder for PAC Codes |

#### 3.2. List Decoding of PAC Codes

Algorithm 2: continuePaths_Unfzn (PAC version) |

## 4. List Decoding versus Sequential Decoding

#### 4.1. Performance Comparison

#### 4.2. Complexity Comparison

## 5. Performance Analysis for PAC Codes

#### 5.1. Sequential Decoding versus ML Decoding

#### 5.2. Weight Distributions and Union Bounds

## 6. PAC Codes with Random Time-Varying Convolutional Precoding

**impulse response**of the convolutional precoder. He furthermore writes in [2] that:

As long as the constraint length of the convolution is sufficiently large, choosing c at random may be an acceptable design practice.

#### 6.1. Random Time-Varying Convolutional Precoding

#### 6.2. Performance of PAC Codes with Random Time-Varying Convolutional Precoding

## 7. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**An example of the polar search tree, reproduced from [2].

**Figure 11.**Performance of PAC codes for some specific realizations of random time-varying convolutional precoding with $\nu =6$, as a function of the list size.

**Figure 12.**Performance of PAC codes for some specific realizations of random time-varying convolutional precoding for $L=128$, as a function of the constraint length.

SNR [dB] | 1.00 | 1.25 | 1.50 | 1.75 | 2.00 | 2.25 |

% of failures | 4.53% | 3.56% | 1.86% | 1.38% | 1.01% | 0.29% |

SNR [dB] | 1.50 | 1.75 | 2.00 | 2.25 | 2.50 | 2.75 | 3.00 |

$L=64$ | 32.1% | 32.2% | 32.5% | 32.3% | 29.4% | 36.7% | 39.6% |

$L=128$ | 50.0% | 51.6% | 54.6% | 53.6% | 58.4% | 60.4% | 63.2% |

$L=256$ | 66.2% | 71.0% | 75.2% | 78.0% | 79.9% | 83.6% | 82.8% |

${\mathit{A}}_{8}$ | ${\mathit{A}}_{12}$ | ${\mathit{A}}_{16}$ | ${\mathit{A}}_{18}$ | ${\mathit{A}}_{20}$ | ${\mathit{A}}_{22}$ | |
---|---|---|---|---|---|---|

Polar code | 48 | 0 | 68,856 | 0 | 897,024 | 0 |

Polar code, CRC8 | 20 | 173 | ⩾7069 | - | - | - |

Reed–Muller | 0 | 0 | 94,488 | 0 | 0 | 0 |

PAC, polar profile | 48 | 0 | 11,032 | 6024 | >${10}^{5}$ | - |

PAC, RM profile | 0 | 0 | 3120 | 2696 | 95,828 | >10${}^{5}$ |

**Table 4.**Number of low-weight codewords in PAC codes for certain specific realizations of random time-varying convolutional precoding, as a function of the constraint length.

${\mathit{A}}_{8}$ | ${\mathit{A}}_{16}$ | ${\mathit{A}}_{18}$ | ${\mathit{A}}_{20}$ | ${\mathit{A}}_{22}$ | |
---|---|---|---|---|---|

Random precoding with $\nu =2$ | 0 | 6424 | 7780 | 142,618 | >10${}^{5}$ |

Arıkan’s PAC code with $\nu =6$ | 0 | 3120 | 2696 | 95,828 | >10${}^{5}$ |

Random precoding with $\nu =6$ | 0 | 2870 | 1526 | 88,250 | >10${}^{5}$ |

Random precoding with $\nu =10$ | 0 | 2969 | 412 | 81,026 | >10${}^{5}$ |

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Yao, H.; Fazeli, A.; Vardy, A. List Decoding of Arıkan’s PAC Codes. *Entropy* **2021**, *23*, 841.
https://doi.org/10.3390/e23070841

**AMA Style**

Yao H, Fazeli A, Vardy A. List Decoding of Arıkan’s PAC Codes. *Entropy*. 2021; 23(7):841.
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**Chicago/Turabian Style**

Yao, Hanwen, Arman Fazeli, and Alexander Vardy. 2021. "List Decoding of Arıkan’s PAC Codes" *Entropy* 23, no. 7: 841.
https://doi.org/10.3390/e23070841