# CRC-Aided Adaptive BP Decoding of PAC Codes

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## Abstract

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## 1. Introduction

- We derive the parity-check matrices of PAC codes suitable for message passing decoding.
- We propose a partial CRC-aided ABP algorithm by designing the joint parity-check matrices of CRC-PAC codes. The ABP algorithm clearly supports the use of partial CRC bits by combining some parity-check constraints of a CRC code and the parity-check matrix of a PAC code.
- We analyze and compare the performance of our proposed algorithm and the conventional CA-List decoding for punctured PAC codes. The performance influence of puncturing is taken into consideration.

## 2. The Structures of PAC Codes and Concatenated CRC-PAC Codes

#### 2.1. The Structures of PAC Codes

#### 2.2. The Structures of CRC-PAC Concatenated Codes

## 3. CRC-Aided ABP Decoding

#### 3.1. The Parity-Check Matrices of CRC-PAC Concatenated Codes

#### 3.2. Decoding Steps of CA-ABP Algorithm

Algorithm 1: ABP algorithm with partial CRC aided, CA-ABP(${T}_{I}$, ${T}_{O}$, $\tilde{L}$) |

#### 3.3. Decoding Complexity

## 4. Punctured PAC Codes

## 5. Simulation Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The performance comparisons of the PAC code and the polar code with $R=96/128$ and $N=128$.

**Figure 2.**The complexity comparisons of CA-List and CA-ABP on the PAC code with $R=40/128$, $N=128$, and $L=24$.

**Figure 3.**The performance of punctured PAC codes decoded by List(32) for different code rates with $N=256$ and $E=240$.

**Figure 9.**The performance comparisons of the punctured PAC code with $R=72/112$, $N=128$, and $E=112$.

**Figure 10.**The performance comparisons of the punctured PAC code with $R=168/224$, $N=256$, and $E=224$.

Algorithms | The Average Computational Complexity |
---|---|

CA-List | $\eta \xb7N\xb7{log}_{2}N+K\xb7(3\xb7\eta -1)\xb7\eta /2$ |

CA-ABP | $((M+\tilde{L})(K-\tilde{L})+(N-1+K-\u03f5)(M+\u03f5)/2)\xb7{T}_{av}$ |

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## Share and Cite

**MDPI and ACS Style**

Zhang, X.; Jiang, M.; Zhu, M.; Liu, K.; Zhao, C.
CRC-Aided Adaptive BP Decoding of PAC Codes. *Entropy* **2022**, *24*, 1170.
https://doi.org/10.3390/e24081170

**AMA Style**

Zhang X, Jiang M, Zhu M, Liu K, Zhao C.
CRC-Aided Adaptive BP Decoding of PAC Codes. *Entropy*. 2022; 24(8):1170.
https://doi.org/10.3390/e24081170

**Chicago/Turabian Style**

Zhang, Xianwen, Ming Jiang, Mingyang Zhu, Kailin Liu, and Chunming Zhao.
2022. "CRC-Aided Adaptive BP Decoding of PAC Codes" *Entropy* 24, no. 8: 1170.
https://doi.org/10.3390/e24081170