{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T13:31:10Z","timestamp":1787146270358,"version":"build-2736575974"},"reference-count":14,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T00:00:00Z","timestamp":1787097600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005372","name":"Capacity Building of Guangxi Key Laboratory of Automatic Detecting Technology and Instruments","doi-asserted-by":"publisher","award":["GuiKeLT2600640021"],"award-info":[{"award-number":["GuiKeLT2600640021"]}],"id":[{"id":"10.13039\/501100005372","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012565","name":"GUAT Special Research Project on the Strategic Development of Distinctive Interdisciplinary Fields","doi-asserted-by":"publisher","award":["TS2024211"],"award-info":[{"award-number":["TS2024211"]}],"id":[{"id":"10.13039\/501100012565","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>BP decoding of BATS codes may stop when no residual batch satisfies the full-row-rank condition. To resume BP decoding, this paper proposes an Important-Packet-Guided Left-Nullspace Rank-Completion (LNRC) method. LNRC first identifies the unrecovered source packet that connects to the largest number of undecoded batches, denotes it as the Important Packet, and uses it as a guidance packet to locate the undecoded batches containing it as repair candidates. For a selected batch with rank deficit one, the destination computes a nonzero left-null vector and selects a local repair coordinate that provides the missing independent direction. The destination sends the corresponding global source-packet index and finite-field coefficient through a reliable reverse feedback-control link, and the source returns the scaled repair packet through a reliable forward repair-data link. The associated completion column increases the target residual transfer-matrix rank by one and makes the batch BP-decodable. Thus, LNRC exploits the column-space structure of the target undecoded batch to select the repair coordinate, rather than selecting the Important Packet solely by the number of connected undecoded batches. Under equal encoding redundancy, simulations show that LNRC achieves a lower packet error rate (PER) compared with conventional Important Packet feedback, with average relative PER reductions of approximately 0.40\u201312.17% across the evaluated settings.<\/jats:p>","DOI":"10.3390\/e28080929","type":"journal-article","created":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T13:14:01Z","timestamp":1787145241000},"page":"929","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["BATS Code Decoding Method Based on Left-Nullspace-Guided Rank-Completion Feedback"],"prefix":"10.3390","volume":"28","author":[{"given":"Juan","family":"Yang","sequence":"first","affiliation":[{"name":"The School of Artificial Intelligence, Guilin University of Aerospace Technology, Guilin 541004, China"},{"name":"Guangxi Key Laboratory of Automatic Detecting Technology and Instruments, Guilin University of Electronic Technology, Guilin 541004, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-6753-8976","authenticated-orcid":false,"given":"Jingjing","family":"Lu","sequence":"additional","affiliation":[{"name":"The School of Artificial Intelligence, Guilin University of Aerospace Technology, Guilin 541004, China"},{"name":"Guangxi Zhuang Autonomous Region Engineering Research Center for Artificial Intelligence and Aerospace Information Scenarios, School of Artificial Intelligence, Guilin University of Aerospace Technology, Guilin 541004, China"},{"name":"University Engineering Research Center of Integrated Aerospace-Terrestrial Information and Control, Guangxi, School of Artificial Intelligence, Guilin University of Aerospace Technology, Guilin 541004, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jianbo","family":"Ji","sequence":"additional","affiliation":[{"name":"The School of Artificial Intelligence, Guilin University of Aerospace Technology, Guilin 541004, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-6640-7713","authenticated-orcid":false,"given":"Tao","family":"Wang","sequence":"additional","affiliation":[{"name":"National Key Laboratory of Wireless Communications (NKLWC), University of Electronic Science and Technology of China (UESTC), Chengdu 610054, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"782","DOI":"10.1109\/TNET.2003.818197","article-title":"An algebraic approach to network coding","volume":"11","author":"Koetter","year":"2003","journal-title":"IEEE\/ACM Trans. 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China"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/8\/929\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T13:17:34Z","timestamp":1787145454000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/8\/929"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,19]]},"references-count":14,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2026,8]]}},"alternative-id":["e28080929"],"URL":"https:\/\/doi.org\/10.3390\/e28080929","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,19]]}}}