Draft:Polarization-adjusted convolutional code
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A polarization-adjusted convolutional code (PAC code) is a class of linear block error-correcting codes that combines a rate-one convolutional precode with the transform used by polar codes. Erdal Arıkan introduced PAC codes in his 2019 Shannon Lecture.[1][2] They were proposed to improve the short-blocklength performance and distance properties of conventional polar codes while retaining a structured encoder.
Construction
[edit]For a block length and dimension , information bits are first placed in a length- vector according to a chosen rate profile; the remaining positions are frozen. A rate-one convolutional transform is then applied, followed by the usual polar transform. Thus the encoder can be viewed as a serial concatenation in which the convolutional transformation is the outer precode and the polar transformation is the inner mapper.[3]
The choice of rate profile affects both the weight distribution and decoding complexity. Profiles based on polar-code reliability and on Reed–Muller code weight rules are commonly studied. Convolutional precoding can reduce the number of low-weight codewords, although the effect depends on the selected profile and precoder.[4]
Decoding
[edit]Arıkan's original proposal used sequential decoding, including the Fano algorithm, on the decoding tree.[1] PAC codes can also be decoded by list-decoding methods adapted from successive-cancellation list decoding of polar codes. Comparative studies have examined the trade-off between sequential-decoding complexity and list-decoding complexity.[5][6]
Performance and applications
[edit]At short block lengths, PAC codes have been reported to improve the block-error performance of conventional polar codes under near-maximum-likelihood decoding.[2][6] Research topics include rate-profile construction, decoder metrics, complexity control, list decoding, and hardware-oriented decoding. Unlike polar and LDPC codes, PAC codes had not been selected as a channel code in a major telecommunications standard as of 2026.
See also
[edit]- 1 2 Arıkan, Erdal (2019). "From sequential decoding to channel polarization and back again". arXiv:1908.09594 [cs.IT].
- 1 2 Gautam, Shashi (2024). "Advanced channel coding schemes for B5G/6G networks: State-of-the-art analysis, research challenges and future directions". International Journal of Communication Systems. doi:10.1002/dac.5855.
- ↑ Rowshan, Mohammad; Viterbo, Emanuele (2021). "On Convolutional Precoding in PAC Codes". 2021 IEEE Globecom Workshops. pp. 1–6. arXiv:2103.12483. doi:10.1109/GCWKSHPS52748.2021.9681987.
- ↑ Rowshan, Mohammad; Yuan, Jinhong (2023). "On the Minimum Weight Codewords of PAC Codes: The Impact of Pre-Transformation". IEEE Journal on Selected Areas in Information Theory. 4: 487–498. doi:10.1109/JSAIT.2023.3312678.
- ↑ Rowshan, Mohammad; Burg, Andreas; Viterbo, Emanuele (2021). "Polarization-Adjusted Convolutional (PAC) Codes: Sequential Decoding vs List Decoding". IEEE Transactions on Vehicular Technology. 70 (2): 1434–1447. doi:10.1109/TVT.2021.3052550.
- 1 2 Yao, Hanwen; Fazeli, Arman; Vardy, Alexander (2021). "List Decoding of Arıkan's PAC Codes". Entropy. 23 (7) 841. doi:10.3390/e23070841. PMC 8303677.
Category:Error detection and correction Category:Coding theory