{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T05:57:38Z","timestamp":1762063058803,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,22]],"date-time":"2022-08-22T00:00:00Z","timestamp":1661126400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Research and Development Program of China","award":["2020YFB1807200","BK20192002"],"award-info":[{"award-number":["2020YFB1807200","BK20192002"]}]},{"name":"Jiangsu Province Basic Research Project","award":["2020YFB1807200","BK20192002"],"award-info":[{"award-number":["2020YFB1807200","BK20192002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Although long polar codes with successive cancellation decoding can asymptotically achieve channel capacity, the performance of short blocklength polar codes is far from optimal. Recently, Ar\u0131kan proposed employing a convolutional pre-transformation before the polarization network, called polarization-adjusted convolutional (PAC) codes. In this paper, we focus on improving the performance of short PAC codes concatenated with a cyclic redundancy check (CRC) outer code, CRC-PAC codes, since error detection capability is essential in practical applications, such as the polar coding scheme for the control channel. We propose an enhanced adaptive belief propagation (ABP) decoding algorithm with the assistance of CRC bits for PAC codes. We also derive joint parity-check matrices of CRC-PAC codes suitable for iterative BP decoding. The proposed CRC-aided ABP (CA-ABP) decoding can effectively improve error performance when partial CRC bits are used in the decoding. Meanwhile, the error detection ability can still be guaranteed by the remaining CRC bits and adaptive decoding parameters. Moreover, compared with the conventional CRC-aided list (CA-List) decoding, our proposed scheme can significantly reduce computational complexity, to achieve a better trade-off between the performance and complexity for short PAC codes.<\/jats:p>","DOI":"10.3390\/e24081170","type":"journal-article","created":{"date-parts":[[2022,8,22]],"date-time":"2022-08-22T21:30:34Z","timestamp":1661203834000},"page":"1170","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["CRC-Aided Adaptive BP Decoding of PAC Codes"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7209-1245","authenticated-orcid":false,"given":"Xianwen","family":"Zhang","sequence":"first","affiliation":[{"name":"The National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ming","family":"Jiang","sequence":"additional","affiliation":[{"name":"The National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China"},{"name":"Purple Mountain Laboratories, Nanjing 211111, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mingyang","family":"Zhu","sequence":"additional","affiliation":[{"name":"The National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kailin","family":"Liu","sequence":"additional","affiliation":[{"name":"The National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chunming","family":"Zhao","sequence":"additional","affiliation":[{"name":"The National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China"},{"name":"Purple Mountain Laboratories, Nanjing 211111, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3051","DOI":"10.1109\/TIT.2009.2021379","article-title":"Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels","volume":"55","author":"Arikan","year":"2009","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1668","DOI":"10.1109\/LCOMM.2012.090312.121501","article-title":"CRC-Aided Decoding of Polar Codes","volume":"16","author":"Niu","year":"2012","journal-title":"IEEE Commun. Lett."},{"key":"ref_3","unstructured":"Arikan, E. (2019). From sequential decoding to channel polarization and back again. arXiv."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Yao, H., Fazeli, A., and Vardy, A. (2020, January 21\u201326). List decoding of Ar\u0131kan\u2019s PAC codes. Proceedings of the IEEE International Symposium on Information Theory, Los Angeles, CA, USA.","DOI":"10.1109\/ISIT44484.2020.9174118"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1434","DOI":"10.1109\/TVT.2021.3052550","article-title":"Emanuele. Polarization-Adjusted Convolutional (PAC) Codes: Sequential Decoding vs List Decoding","volume":"70","author":"Rowshan","year":"2021","journal-title":"IEEE Trans. Veh. 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