{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:08:48Z","timestamp":1760058528560,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,9]],"date-time":"2025-04-09T00:00:00Z","timestamp":1744156800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Key Science and Technology Research Project of Henan Province of China","award":["222102210053","21A510003","231100220400","231100220300","ZZU-CISS-2024003"],"award-info":[{"award-number":["222102210053","21A510003","231100220400","231100220300","ZZU-CISS-2024003"]}]},{"name":"Key Scientific Research Project in Colleges and Universities of Henan Province of China","award":["222102210053","21A510003","231100220400","231100220300","ZZU-CISS-2024003"],"award-info":[{"award-number":["222102210053","21A510003","231100220400","231100220300","ZZU-CISS-2024003"]}]},{"name":"Major Science and Technology Projects of Longmen Laboratory","award":["222102210053","21A510003","231100220400","231100220300","ZZU-CISS-2024003"],"award-info":[{"award-number":["222102210053","21A510003","231100220400","231100220300","ZZU-CISS-2024003"]}]},{"name":"Open Fund of Intelligent Group System Engineering Research Center of the Ministry of Education","award":["222102210053","21A510003","231100220400","231100220300","ZZU-CISS-2024003"],"award-info":[{"award-number":["222102210053","21A510003","231100220400","231100220300","ZZU-CISS-2024003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The Alternating Direction Method of Multipliers (ADMM) has proven to be an efficient approach for implementing linear programming (LP) decoding of low-density parity-check (LDPC) codes. By introducing penalty terms into the LP decoding model\u2019s objective function, ADMM-based variable node penalized decoding effectively mitigates non-integral solutions, thereby improving frame error rate (FER) performance, especially in the low signal-to-noise ratio (SNR) region. In this paper, we leverage the ADMM framework to derive explicit iterative steps for solving the LP decoding problem for LDPC codes with penalty functions. To further enhance decoding efficiency and accuracy, We propose an LDPC code decoding algorithm based on the symmetric ADMM (S-ADMM). We also establish some contraction properties satisfied by the iterative sequence of the algorithm. Through simulation experiments, we evaluate the proposed S-ADMM decoder using three standard LDPC codes and three representative fifth-generation (5G) codes. The results show that the S-ADMM decoder consistently outperforms conventional ADMM penalized decoders, offering significant improvements in decoding performance.<\/jats:p>","DOI":"10.3390\/e27040404","type":"journal-article","created":{"date-parts":[[2025,4,9]],"date-time":"2025-04-09T12:05:56Z","timestamp":1744200356000},"page":"404","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Low-Density Parity-Check Decoding Algorithm Based on Symmetric Alternating Direction Method of Multipliers"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7642-7607","authenticated-orcid":false,"given":"Ji","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471000, China"},{"name":"Intelligent System Science and Technology Innovation Center, Longmen Laboratory, Luoyang 471000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anmin","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ying","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Baofeng","family":"Ji","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471000, China"},{"name":"Intelligent System Science and Technology Innovation Center, Longmen Laboratory, Luoyang 471000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3779-4505","authenticated-orcid":false,"given":"Huaan","family":"Li","sequence":"additional","affiliation":[{"name":"College of Physics and Telecommunication Engineering, Zhoukou Normal University, Zhoukou 466001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hengzhou","family":"Xu","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":[[2025,4,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"954","DOI":"10.1109\/TIT.2004.842696","article-title":"Using linear programming to Decode Binary linear codes","volume":"51","author":"Feldman","year":"2005","journal-title":"IEEE Trans. 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