{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:12:20Z","timestamp":1760058740409,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,4,27]],"date-time":"2025-04-27T00:00:00Z","timestamp":1745712000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["62172457","11971311","61801527","12026230","12026231","252102210230","252102210236","242102210200","22A520002","24A510003"],"award-info":[{"award-number":["62172457","11971311","61801527","12026230","12026231","252102210230","252102210236","242102210200","22A520002","24A510003"]}]},{"name":"TianYuan Special Funds of the National Natural Science Foundation of China","award":["62172457","11971311","61801527","12026230","12026231","252102210230","252102210236","242102210200","22A520002","24A510003"],"award-info":[{"award-number":["62172457","11971311","61801527","12026230","12026231","252102210230","252102210236","242102210200","22A520002","24A510003"]}]},{"name":"Development Project of Henan Provincial Department of Science and Technology","award":["62172457","11971311","61801527","12026230","12026231","252102210230","252102210236","242102210200","22A520002","24A510003"],"award-info":[{"award-number":["62172457","11971311","61801527","12026230","12026231","252102210230","252102210236","242102210200","22A520002","24A510003"]}]},{"name":"Key Scientific Research Project in Colleges and Universities of Henan Province","award":["62172457","11971311","61801527","12026230","12026231","252102210230","252102210236","242102210200","22A520002","24A510003"],"award-info":[{"award-number":["62172457","11971311","61801527","12026230","12026231","252102210230","252102210236","242102210200","22A520002","24A510003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we investigate the cycle structure inherent in the Tanner graphs of low-density parity-check (LDPC) codes constructed from balanced incomplete block designs (BIBDs). We begin by delineating the incidence structure of BIBDs and propose a methodology for constructing LDPC codes based on these designs. By analyzing the incidence relations between points and blocks within a BIBD, we prove that the resulting LDPC codes possess a girth of 6. Subsequently, we provide a detailed analysis of the cycle structure of the constructed LDPC codes and introduce a systematic approach for enumerating their short cycles. Using this method, we determine the exact numbers of cycles of lengths 6 and 8. Simulation results demonstrate that the constructed LDPC codes exhibit excellent performance.<\/jats:p>","DOI":"10.3390\/e27050476","type":"journal-article","created":{"date-parts":[[2025,4,29]],"date-time":"2025-04-29T09:00:39Z","timestamp":1745917239000},"page":"476","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["LDPC Codes on Balanced Incomplete Block Designs: Construction, Girth, and Cycle Structure Analysis"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9403-4460","authenticated-orcid":false,"given":"Hengzhou","family":"Xu","sequence":"first","affiliation":[{"name":"School of Computer, Henan University of Engineering, Zhengzhou 451191, China"},{"name":"School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaodong","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mengmeng","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Computer, Henan University of Engineering, Zhengzhou 451191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haipeng","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Computer, Henan University of Engineering, Zhengzhou 451191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2899-5404","authenticated-orcid":false,"given":"Hai","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Computer, Henan University of Engineering, Zhengzhou 451191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1109\/TIT.1981.1056404","article-title":"A recursive approach to low complexity codes","volume":"27","author":"Tanner","year":"1981","journal-title":"IEEE Trans. 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