{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:59:43Z","timestamp":1760151583694,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,4,11]],"date-time":"2022-04-11T00:00:00Z","timestamp":1649635200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61372072"],"award-info":[{"award-number":["61372072"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100013313","name":"Overseas Expertise Introduction Project for Discipline Innovation","doi-asserted-by":"publisher","award":["B08038"],"award-info":[{"award-number":["B08038"]}],"id":[{"id":"10.13039\/501100013313","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Belief propagation (BP) decoding for polar codes has been extensively studied because of its inherent parallelism. However, its performance remains inferior to that of successive cancellation list decoding (SCL) due to the structure of the decoding graph. To improve the block error rate (BLER) performance, the BP correction (BPC) decoding, a post-processing scheme that corrects prior knowledge of the identified code bit, improves convergence by executing additional iterations on the failed BP decoder. Moreover, the BPC decoder demonstrates a better decoding performance than the BP-based bit-flipping decoder. Nevertheless, the additional decoding attempts lead to increased latency. In this article, a modified BPC decoder is proposed to reduce the number of decoding attempts by redefining the correction rules. A new metric is designed to effectively identify the corrected location. Numerical results show that the proposed modified BPC decoder achieves a slight improvement in BLER compared with the original BPC, with a dramatic reduction in average complexity. Furthermore, a higher-order version, named MBPC-\u03a9, is extended to further improve the performance, where the \u03a9 is the maximum correction order. Numerical results show that the higher-order modified BPC achieves a similar BLER performance to existing multiple bit-flipping BP decoders but has around half the latency overhead. In addition, the proposed MBPC-2 decoder performs better than the cyclic redundancy check-aided SCL (CA-SCL) decoder with list size 4 and is slightly worse than the CA-SCL with list size 8 in high signal-to-noise ratio (SNR) regions but with significant decoding latency reduction.<\/jats:p>","DOI":"10.3390\/e24040534","type":"journal-article","created":{"date-parts":[[2022,4,12]],"date-time":"2022-04-12T00:23:11Z","timestamp":1649722991000},"page":"534","update-policy":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Higher-Order Belief Propagation Correction Decoder for Polar Codes"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/linproxy.fan.workers.dev:443\/https\/orcid.org\/0000-0001-6591-6894","authenticated-orcid":false,"given":"Meng","family":"Zhang","sequence":"first","affiliation":[{"name":"The State Key Laboratory of Integrated Services Networks, Xidian University, Xi\u2019an 710071, China"}]},{"ORCID":"https:\/\/linproxy.fan.workers.dev:443\/https\/orcid.org\/0000-0002-4345-5764","authenticated-orcid":false,"given":"Zhuo","family":"Li","sequence":"additional","affiliation":[{"name":"The State Key Laboratory of Integrated Services Networks, Xidian University, Xi\u2019an 710071, China"}]},{"given":"Lijuan","family":"Xing","sequence":"additional","affiliation":[{"name":"The State Key Laboratory of Integrated Services Networks, Xidian University, Xi\u2019an 710071, China"}]},{"given":"Xin","family":"Liao","sequence":"additional","affiliation":[{"name":"The State Key Laboratory of Integrated Services Networks, Xidian University, Xi\u2019an 710071, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,11]]},"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","year":"2009","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","unstructured":"(2021, November 10). Final Report of 3GPP TSG RAN WG1 #87 v1.0.0; Reno, NV, USA. November 2016. Available online: https:\/\/linproxy.fan.workers.dev:443\/https\/www.3gpp.org\/ftp\/tsg_ran\/WG1_RL1\/TSGR1_87\/Report\/Final_Minutes_report_RAN1%2387_v100.zip."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2213","DOI":"10.1109\/TIT.2015.2410251","article-title":"List Decoding of Polar Codes","volume":"61","author":"Tal","year":"2015","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_4","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_5","unstructured":"Ar\u0131kan, E. (2010, January 11\u201314). Polar Codes: A Pipelined Implementation. Proceedings of the 4th International Symposium on Broadband Communication (ISBC 2010), Melaka, Malaysia."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1109\/LCOMM.2008.080017","article-title":"A performance comparison of polar codes and reed-muller codes","volume":"12","year":"2008","journal-title":"IEEE Commun. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Elkelesh, A., Ebada, M., Cammerer, S., and Brink, S.T. (2017, January 28\u201331). Mitigating clipping effects on error floors under belief propagation decoding of polar codes. Proceedings of the 2017 International Symposium on Wireless Communication Systems (ISWCS), Bologna, Italy.","DOI":"10.1109\/ISWCS.2017.8108145"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Doan, N., and Hashemi, S.A. (2018, January 9\u201313). On the decoding of polar codes on permuted factor graphs. Proceedings of the 2018 IEEE Global Communications Conference (GLOBECOM), Abu Dhabi, United Arab Emirates.","DOI":"10.1109\/GLOCOM.2018.8647308"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"17632","DOI":"10.1109\/ACCESS.2020.2968100","article-title":"Belief propagation with permutated graphs of polar codes","volume":"8","author":"Li","year":"2020","journal-title":"IEEE Access"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ranasinghe, V., and Rajatheva, N. (2020, January 7\u201311). Partially permuted multi-trellis belief propagation for polar codes. Proceedings of the 2020 IEEE International Conference on Communications (ICC), Dublin, Ireland.","DOI":"10.1109\/ICC40277.2020.9149228"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1536","DOI":"10.1109\/LCOMM.2018.2850772","article-title":"Belief propagation list decoding of polar codes","volume":"22","author":"Elkelesh","year":"2018","journal-title":"IEEE Commun. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"5657","DOI":"10.1109\/TVT.2020.2979334","article-title":"Efficient belief propagation polar decoder with loop simplification based factor graphs","volume":"69","author":"Ren","year":"2020","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Sun, S., Cho, S.G., and Zhang, Z. (2017, January 4\u20138). Post-processing methods for improving coding gain in belief propagation decoding of polar codes. Proceedings of the IEEE Global Communications Conference (GLOBECOM), Singapore.","DOI":"10.1109\/GLOCOM.2017.8254247"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1285","DOI":"10.1109\/LCOMM.2019.2918535","article-title":"Noise-aided belief propagationlist decoding of polar codes","volume":"23","author":"Gazi","year":"2019","journal-title":"IEEE Commun. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"10937","DOI":"10.1109\/ACCESS.2019.2891951","article-title":"Belief propagation bit-flip decoder for polar codes","volume":"7","author":"Yu","year":"2019","journal-title":"IEEE Access"},{"key":"ref_16","first-page":"901","article-title":"Enhanced belief propagation decoder for 5G polar codes with bit-flipping","volume":"67","author":"Shen","year":"2020","journal-title":"IEEE Trans. Circuits Syst. II Exp. Briefs"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6699","DOI":"10.1109\/TCOMM.2020.3017656","article-title":"Improved Belief Propagation Polar Decoders With Bit-Flipping Algorithms","volume":"68","author":"Shen","year":"2020","journal-title":"IEEE Trans. Commun."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"83710","DOI":"10.1109\/ACCESS.2020.2988878","article-title":"Belief Propagation Decoder With Multiple Bit-Flipping Sets and Stopping Criteria for Polar Codes","volume":"8","author":"Zhang","year":"2020","journal-title":"IEEE Access"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1109\/TSP.2020.3040897","article-title":"Convolutional neural network-aided tree-based bit-flipping framework for polar decoder using imitation learning","volume":"69","author":"Teng","year":"2020","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_20","unstructured":"Yang, Y., Yin, C., Jan, Q., Hu, Y., Pan, Z., Liu, N., and You, X. (2020, January 21\u201323). Noise-aided belief propagation list bit-flip decoder for polar codes. Proceedings of the 2020 International Conference on Wireless Communications and Signal Processing(WCSP), Wuhan, China."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2468","DOI":"10.1109\/LCOMM.2021.3086479","article-title":"A Novel Post-Processing Method for Belief Propagation List Decoding of Polar Codes","volume":"25","author":"Feng","year":"2021","journal-title":"IEEE Commun. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3161","DOI":"10.1109\/LCOMM.2021.3104152","article-title":"An Enhanced Belief Propagation Decoder for Polar Codes","volume":"25","author":"Zhang","year":"2021","journal-title":"IEEE Commun. Lett."},{"key":"ref_23","unstructured":"Hussami, N., Korada, S.B., and Urbanke, R. (July, January 28). Performance of polar codes for channel and source coding. Proceedings of the IEEE International Symposium on Information Theory (ISIT), Seoul, Korea."},{"key":"ref_24","unstructured":"Eslami, A., and Pishro-Nik, H. (October, January 29). On bit error rate performance of polar codes in finite regime. Proceedings of the 48th Annual Allerton Conference on Communication, Control, and Computing(Allerton), Monticello, IL, USA."},{"key":"ref_25","unstructured":"Eslami, A., and Pishro-Nik, H. (August, January 31). A practical approach to polar codes. Proceedings of the IEEE International Symposium on Information Theory (ISIT), Saint Petersburg, Russia."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Chandesris, L., Savin, V., and Declercq, D. (2016, January 4\u20138). An improved SCFlip decoder for polar codes. Proceedings of the 2016 IEEE Global Communications Conference (GLOBECOM), Washington, DC, USA.","DOI":"10.1109\/GLOCOM.2016.7841594"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"5756","DOI":"10.1109\/TSP.2017.2740204","article-title":"Fast and Flexible Successive-Cancellation List Decoders for Polar Codes","volume":"65","author":"Hashemi","year":"2017","journal-title":"IEEE Trans. Signal Process."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/www.mdpi.com\/1099-4300\/24\/4\/534\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:51:36Z","timestamp":1760136696000},"score":1,"resource":{"primary":{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/www.mdpi.com\/1099-4300\/24\/4\/534"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,11]]},"references-count":27,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["e24040534"],"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.3390\/e24040534","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,4,11]]}}}