#
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

- Arıkan, E. Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels. IEEE Trans. Inf. Theory
**2009**, 55, 3051–3073. [Google Scholar] [CrossRef] - Arıkan, E. From sequential decoding to channel polarization and back again. arXiv
**2019**, arXiv:1908.09594. [Google Scholar] - Polyanskiy, Y.; Poor, H.V.; Verdu, S. Channel coding rate in the finite blocklength regime. IEEE Trans. Inf. Theory
**2010**, 56, 2307–2359. [Google Scholar] [CrossRef] - Tal, I.; Vardy, A. List decoding of polar codes. IEEE Trans. Inf. Theory
**2015**, 61, 2213–2226. [Google Scholar] [CrossRef] - Li, B.; Shen, H.; Tse, D. An adaptive successive cancellation list decoder for polar codes with cyclic redundancy check. IEEE Commun. Lett.
**2012**, 16, 2044–2047. [Google Scholar] [CrossRef] [Green Version] - Miloslavskaya, V.; Trifonov, P. Sequential decoding of polar codes. IEEE Commun. Lett.
**2014**, 16, 1127–1130. [Google Scholar] [CrossRef] - Niu, K.; Chen, K. CRC-aided decoding of polar codes. IEEE Commun. Lett.
**2012**, 16, 1668–1671. [Google Scholar] [CrossRef] - 3GPP Technical Specification Group Radio Access Network, “Multiplexing and channel coding” Release 16, 3GPP TS 38.212 V16.3.0. November 2020. Available online: https://www.etsi.org/deliver/etsi_ts/138200_138299/138212/16.03.00_60/ts_138212v160300p.pdf (accessed on 27 June 2021).
- Erseghe, T. Coding in the finite-blocklength regime: Bounds based on Laplace integrals and their asymptotic approximations. IEEE Trans. Inf. Theory
**2016**, 62, 6854–6883. [Google Scholar] [CrossRef] - Fano, R. A heuristic discussion of probabilistic decoding. IEEE Trans. Inf. Theory
**1963**, 9, 64–74. [Google Scholar] [CrossRef] - Gallager, R.G. Information Theory and Reliable Communication; Wiley: New York, NY, USA, 1968. [Google Scholar]
- Trifonov, P. A score function for sequential decoding of polar codes. In Proceedings of the IEEE International Symposium on Information Theory, Vail, CO, USA, 21–26 June 2018; pp. 1470–1474. [Google Scholar]
- Fazeli, A.; Hassani, H.; Mondelli, M.; Vardy, A. Binary linear codes with optimal scaling: Polar codes with large kernels. IEEE Trans. Inf. Theory
**2020**. [Google Scholar] [CrossRef] - Fazeli, A.; Vardy, A. On the scaling exponent of binary polarization kernels. In Proceedings of the Allerton Conference Communication, Control, and Computing, Monticello, IL, USA, 30 September–3 October 2014; pp. 797–804. [Google Scholar]
- Korada, S.B.; Şaşoğlu, E.; Urbanke, R. Polar codes: Characterization of exponent, bounds, and constructions. IEEE Trans. Inf. Theory
**2010**, 56, 6253–6264. [Google Scholar] [CrossRef] [Green Version] - Moskovskaya, E.; Trifonov, P. Design of BCH polarization kernels with reduced processing complexity. IEEE Commun. Lett.
**2020**, 24, 1383–1386. [Google Scholar] [CrossRef] - Trifonov, P. On construction of polar subcodes with large kernels. In Proceedings of the IEEE International Symposium on Information Theory, Paris, France, 7–12 July 2019; pp. 1932–1936. [Google Scholar]
- Trifonov, P. Trellis-based decoding techniques for polar codes with large kernels. In Proceedings of the IEEE Information Theory Workshop, Visby, Sweden, 25–28 August 2019; pp. 1–5. [Google Scholar]
- Trofimiuk, G.; Trifonov, P. Efficient decoding of polar codes with some 16×16 kernels. In Proceedings of the IEEE Information Theory Workshop, Guangzhou, China, 25–29 November 2018; pp. 11–15. [Google Scholar]
- Trofimiuk, G.; Trifonov, P. Reduced complexity window processing of binary polarization kernels. In Proceedings of the IEEE International Symposium on Information Theory, Paris, France, 7–12 July 2019; pp. 1412–1416. [Google Scholar]
- Yao, H.; Fazeli, A.; Vardy, A. Explicit polar codes with small scaling exponent. In Proceedings of the IEEE International Symposium on Information Theory, Paris, France, 7–12 July 2019; pp. 1757–1761. [Google Scholar]
- Morozov, R.; Trifonov, P. Successive and two-stage systematic encoding of polar subcodes. IEEE Wireless Commun. Lett.
**2019**, 8, 877–880. [Google Scholar] [CrossRef] - Trifonov, P. Star polar subcodes. In Proceedings of the IEEE Wireless Communications and Networking Conference, San Francisco, CA, USA, 17–21 March 2017; pp. 1–6. [Google Scholar]
- Trifonov, P. Randomized polar subcodes with optimized error coefficient. IEEE Trans. Commun.
**2020**, 68, 6714–6722. [Google Scholar] [CrossRef] - Trifonov, P.; Miloslavskaya, V. Polar codes with dynamic frozen symbols and their decoding by directed search. In Proceedings of the IEEE Information Theory Workshop, Sevilla, Spain, 9–13 September 2013; pp. 1–5. [Google Scholar]
- Trifonov, P.; Miloslavskaya, V. Polar subcodes. IEEE J. Sel. Areas Commun.
**2016**, 34, 254–266. [Google Scholar] [CrossRef] [Green Version] - Trifonov, P.; Trofimiuk, G. A randomized construction of polar subcodes. In Proceedings of the IEEE International Symposium on Information Theory, Aachen, Germany, 25–30 June 2017; pp. 1863–1867. [Google Scholar]
- Ferris, A.J.; Poulin, D. Branching MERA codes: A natural extension of polar codes. arXiv
**2013**, arXiv:1312.4575. [Google Scholar] - Ferris, A.J.; Hirche, D.C.; Poulin, D. Convolutional polar codes. arXiv
**2017**, arXiv:1704.00715. [Google Scholar] - Morozov, R. Convolutional polar kernels. IEEE Trans. Commun.
**2020**, 68, 7352–7361. [Google Scholar] [CrossRef] - Morozov, R.; Trifonov, P. On distance properties of convolutional polar codes. IEEE Trans. Commun.
**2019**, 67, 4585–4592. [Google Scholar] [CrossRef] - Abbe, E.; Ye, M. Reed-Muller codes polarize. IEEE Trans. Inf. Theory
**2020**, 66, 7311–7332. [Google Scholar] [CrossRef] - Li, B.; Shen, H.; Tse, D. RM-polar codes. arXiv
**2014**, arXiv:1407.5483. [Google Scholar] - Mondelli, M.; Hassani, S.H.; Urbanke, R.L. From polar to Reed–Muller codes: A technique to improve the finite-length performance. IEEE Trans. Commun.
**2020**, 62, 3084–3091. [Google Scholar] [CrossRef] [Green Version] - Ye, M.; Abbe, E. Recursive projection-aggregation decoding of Reed–Muller codes. arXiv
**2019**, arXiv:1902.01470. [Google Scholar] - Coşkun, M.C.; Neu, J.; Pfister, H.D. Successive cancellation inactivation decoding for modified Reed–Muller and eBCH codes. In Proceedings of the IEEE International Symposium on Information Theory, Los Angeles, CA, USA, 21–26 June 2020; pp. 437–442. [Google Scholar]
- Miloslavskaya, V.; Vucetic, B. Design of short polar codes for SCL decoding. IEEE Trans. Commun.
**2021**, 68, 6657–6668. [Google Scholar] [CrossRef] - Yuan, P.; Prinz, T.; Böcherer, G.; Iscan, O.; Boehnke, R.; Xu, W. Polar code construction for list decoding. In Proceedings of the 12th International ITG Conference on Systems, Communications and Coding, Rostock, Germany, 11–14 February 2019; pp. 1–6. [Google Scholar]
- Fazeli, A.; Tian, K.; Vardy, A. Viterbi-aided successive-cancellation decoding of polar codes. In Proceedings of the IEEE Global Communications Conference, Singapore, 4–8 December 2017; pp. 1–6. [Google Scholar]
- Fazeli, A.; Vardy, A.; Yao, H. Convolutional decoding of polar codes. In Proceedings of the IEEE International Symposium on Information Theory, Paris, France, 7–12 July 2019; pp. 1397–1401. [Google Scholar]
- Arıkan, E. Systematic encoding and shortening of PAC Codes. Entropy
**2020**, 22, 1301. [Google Scholar] [CrossRef] - Li, B.; Zhang, H.; Gu, J. On pre-transformed polar codes. arXiv
**2019**, arXiv:1912.06359. [Google Scholar] - Mishra, S.K.; Kim, K.C. Selectively precoded polar codes. arXiv
**2020**, arXiv:2011.04930. [Google Scholar] - Moradi, M.; Mozammel, A.; Qin, K.; Arıkan, E. Performance and complexity of sequential decoding of PAC codes. arXiv
**2020**, arXiv:2012.04990v1. [Google Scholar] - Mozammel, A. Hardware implementation of Fano decoder for PAC codes. arXiv
**2020**, arXiv:2011.09819. [Google Scholar] - Tonnellier, T.; Gross, W.J. On systematic polarization-adjusted convolutional (PAC) codes. arXiv
**2020**, arXiv:2011.03177. [Google Scholar] - Wang, L.; Jiang, M.; Zhao, C.; Li, Z. Genetic optimization of short block-length PAC codes for high capacity PHz communications. In Proceedings of the International Conference on Optoelectronic and Microelectronic Technology and Application, Nanjing, China, 17–19 October 2020; Volume 11617. [Google Scholar]
- Rowshan, M.; Burg, A.; Viterbo, E. Polarization-adjusted convolutional (PAC) codes: Fano decoding vs. list decoding. arXiv
**2020**, arXiv:2002.06805v1. [Google Scholar] - Rowshan, M.; Viterbo, E. List Viterbi decoding of PAC codes. arXiv
**2020**, arXiv:2007.05353. [Google Scholar] - Yao, H.; Fazeli, A.; Vardy, A. List decoding of Arıkan’s PAC codes. In Proceedings of the IEEE International Symposium on Information Theory, Los Angeles, CA, USA, 21–26 June 2020; pp. 443–448. [Google Scholar]
- Vazquez-Vilar, G.; Fabregas, A.G.; Koch, T.; Lancho, A. Saddlepoint approximation of the error probability of binary hypothesis testing. In Proceedings of the IEEE International Symposium on Information Theory, Vail, CO, USA, 17–22 June 2018; pp. 2306–2310. [Google Scholar]
- Coşkun, M.C.; Durisi, G.; Jerkovits, T.; Liva, G.; Ryan, W.; Stein, B.; Steiner, F. Efficient error-correcting codes in the short blocklength regime. Phys. Commun.
**2019**, 34, 66–79. [Google Scholar] [CrossRef] [Green Version] - Goldin, D.; Burshtein, D. Performance bounds of concatenated polar coding schemes. IEEE Trans. Inf. Theory
**2019**, 65, 7131–7148. [Google Scholar] [CrossRef] [Green Version] - Sarkis, G.; Giard, P.; Vardy, A.; Thibeault, C.; Gross, W.J. Increasing the speed of polar list decoders. In Proceedings of the IEEE Workshop on Signal Processing Systems (SiPS), Belfast, UK, 20–22 October 2014; pp. 1–6. [Google Scholar]
- Sarkis, G.; Giard, P.; Vardy, A.; Thibeault, C.; Gross, W.J. Fast list decoders for polar codes. IEEE J. Sel. Areas Commun.
**2016**, 34, 318–328. [Google Scholar] [CrossRef] [Green Version] - Balatsoukas-Stimming, A. Private Communication, August 2019.
- Sugino, M.; Ienaga, Y.; Tokura, N.; Kasami, T. Weight distribution of (128,64) Reed–Muller code. IEEE Trans. Inf. Theory
**1971**, 17, 627–628. [Google Scholar] [CrossRef] - Yao, H.; Fazeli, A.; Vardy, A. A deterministic algorithm for computing the weight distribution of polar codes. arXiv
**2021**, arXiv:2102.07362. [Google Scholar] - Sason, I.; Shamai, S. Performance analysis of linear codes under maximum-likelihood decoding: A tutorial. Found. Trends Commun. Inf. Theory
**2006**, 3, 1–225. [Google Scholar] [CrossRef] - Zhu, H.; Cao, Z.; Zhao, Y.; Li, D.; Yang, Y.; Wang, Y.; Guo, Z. Fast list decoders for polarization-adjusted convolutional (PAC) codes. arXiv
**2020**, arXiv:20122.09425. [Google Scholar]

**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.
https://doi.org/10.3390/e23070841

**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