{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T16:03:57Z","timestamp":1774541037853,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,12]],"date-time":"2025-06-12T00:00:00Z","timestamp":1749686400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key R&amp;D Program of China","award":["2023YFA1009601"],"award-info":[{"award-number":["2023YFA1009601"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we propose the threshold successive cancellation flip (Th-SCF) decoding algorithm for polar codes, which enhances the performance of the SC decoder while maintaining low complexity. Theoretical analysis reveals that Th-SCF asymptotically delays the first error position (FEP, the first part where the SC decoder fails) with probability 1, ensuring high decoding performance. Simulation results show that the Th-SCF algorithm achieves performance comparable to the dynamic SC flip (D-SCF) algorithm, but with a reduction in complexity by eliminating the need for sorting operations. A key contribution of this work is the rigorous theoretical framework supporting the Th-SCF algorithm, distinguishing it from existing SC flip (SCF) decoding methods. This theoretical foundation not only explains the performance improvements but also provides insights into the underlying mechanisms of flipping. The proposed Th-SCF algorithm demonstrates strong performance across a wide range of code lengths and rates, and its performance remains stable within a certain threshold range, indicating its practical applicability in real-world communication systems. These results offer valuable perspectives for the design of efficient flip decoding strategies in 5G and future networks.<\/jats:p>","DOI":"10.3390\/e27060626","type":"journal-article","created":{"date-parts":[[2025,6,12]],"date-time":"2025-06-12T11:47:07Z","timestamp":1749728827000},"page":"626","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Threshold Successive Cancellation Flip Decoding Algorithm for Polar Codes: Design and Performance"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-0419-3876","authenticated-orcid":false,"given":"Zhicheng","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-2382-9918","authenticated-orcid":false,"given":"Liuquan","family":"Yao","sequence":"additional","affiliation":[{"name":"University of Chinese Academy of Sciences, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-8137-8918","authenticated-orcid":false,"given":"Shuai","family":"Yuan","sequence":"additional","affiliation":[{"name":"University of Chinese Academy of Sciences, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guiying","family":"Yan","sequence":"additional","affiliation":[{"name":"University of Chinese Academy of Sciences, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1169-7627","authenticated-orcid":false,"given":"Zhiming","family":"Ma","sequence":"additional","affiliation":[{"name":"University of Chinese Academy of Sciences, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3707-982X","authenticated-orcid":false,"given":"Yuting","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,12]]},"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\u2014Achieving Codes for Symmetric Binary-Input Memoryless Channels","volume":"55","year":"2009","journal-title":"IEEE Trans. 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