[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85090-en":3,"doc-seo-85090-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85090,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Learning LDPC Codes with Quantized Density Evolution over Relaxed Protographs","The document addresses designing low-density parity-check (LDPC) codes matched to a specified iterative decoder, focusing on the parity-check matrix/protograph as a hard combinatorial optimization problem. It reviews prior approaches using simulation, density evolution (DE), EXIT analysis, and costly decoder-in-the-loop gradient descent with noisy Monte Carlo estimates and relaxed-to-integer mismatch. A deterministic GD-based framework optimizes long protograph LDPC codes over relaxed protograph probabilities, using DE-BER loss evaluable directly for relaxed forms, with ensemble justification and faster, more reliable convergence. Experiments with min-sum decoding show improved performance over 5G LDPC codes of identical protograph dimensions.","Learning LDPC codes with quantized density evolution over relaxed protographs  \nGennady Shutkov, Dmitry Artemasov, Alexey Frolov, Pavel Rybin, Kirill Andreev Center for Next Generation Wireless and IoT  \nSkolkovo Institute of Science and Technology  \nMoscow, Russia  \n{g.shutkov, d.artemasov, al.frolov, p.rybin, [k.andreev](k.andreev}@skoltech.ru)[}](k.andreev}@skoltech.ru)[@skoltech.ru](k.andreev}@skoltech.ru)  \narXiv :2607 .08484v 1 [ cs .IT] 9 Jul 2026  \nAbstract—We consider the problem of designing low-density parity-check (LDPC) codes for a given iterative decoder. Although numerous methods are available for evaluating LDPC code performance, including direct simulation, density evolution (DE) and EXIT-chart analysis, the selection of a paritycheck matrix remains a challenging combinatorial optimization problem. Existing design approaches often rely on populationbased search, random mutations, genetic algorithms, or their variants, which require careful parameter tuning and can be computationally expensive. Recent gradient descend (GD)-based code-design methods optimize relaxed parity-check matrices by differentiating through decoder simulations. However, such decoder-in-the-loop strategies rely on noisy Monte Carlo estimates, require line search over soft matrix representations, and remain costly for long LDPC codes. In addition, although the optimization is performed over a relaxed representation, the loss function is typically evaluated only at integer-valued parity-check matrices. In this work, we continue this line of research and focus on the design of long protograph-based LDPC codes. We propose a deterministic GD-based framework that operates directly with a relaxed protograph representation, where each protograph entry is interpreted as the probability that the corresponding element is equal to one. The proposed loss function is based on DE bit error rate (BER) performance and, importantly, can be evaluated directly for relaxed protographs. To justify this relaxation, we associate the relaxed representation with an ensemble of binary protographs and show that the proposed relaxed DE yields the ensemble-averaged DE performance. The resulting optimization procedure is fully autonomous and can employ standard GD optimization. Due to deterministic DE evaluation and informative gradient magnitudes, the proposed approach provides fast and reliable convergence. Numerical experiments for the min-sum decoder demonstrate that the optimized protographs outperform 5G LDPC codes with the same protograph dimensions.  \nIndex Terms—LDPC codes, protographs, density evolution, gradient descent, code optimization.  \nI. INTRODUCTION  \nLow-density parity-check (LDPC) codes are among the most successful classes of forward-error-correction codes. Since the original works of Gallager and Tanner [1], [2], the construction of LDPC codes has become a long-standing and practically important research problem. Under beliefpropagation (BP), min-sum, and related message-passing decoders, the performance of an LDPC code is determined not only by its rate and block length, but also by the detailed structure of the Tanner graph. Consequently, LDPC code design is, to a large extent, a graph-design problem.  \nClassical LDPC design methods address this problem through a combination of asymptotic analysis and finite-length graph construction. A large body of work has developed methods for evaluating the performance of LDPC codes with different graph structures, decoding rules, and channel models. Density evolution (DE) predicts iterative-decoding thresholdsand provides a principled way to optimize irregular ensembles [3], [4], while extrinsic information transfer (EXIT) charts give a graphical tool for analyzing convergence of iterative decoders [5] . For protograph-based ensembles, protograph EXIT (P-EXIT) analysis adapts this idea to the base graph and enables efficient threshold optimization of structured LDPC codes [6] . At finite lengths, graph-con","cbCaibP1pbI6dTuT","https://ap.wps.com/l/cbCaibP1pbI6dTuT","pdf",583433,2,1,9,"English","en",105,"# Introduction\n## LDPC performance depends on Tanner graph structure\n## Classical tools for evaluation and construction\n## Core synthesis challenge: decoder- and constraint-aware design\n## Limitations of discrete and population-based optimization approaches","[{\"question\":\"What optimization challenge does the document address in LDPC code design?\",\"answer\":\"Selecting a parity-check matrix or protograph best suited to a prescribed iterative decoder is treated as a difficult combinatorial optimization problem, especially under practical constraints like finite iterations and quantized messages.\"},{\"question\":\"Why are existing gradient-descent-based code-design methods costly or limited?\",\"answer\":\"Decoder-in-the-loop strategies require noisy Monte Carlo estimates, line search over soft matrix representations, and often evaluate loss only at integer-valued parity-check matrices, making them expensive for long LDPC codes.\"},{\"question\":\"How does the proposed method optimize relaxed protograph representations?\",\"answer\":\"Each relaxed protograph entry is interpreted as the probability that the corresponding element equals one, and the loss function is built from DE bit error rate (BER) performance that can be evaluated directly for relaxed protographs. The work also connects the relaxation to an ensemble interpretation and uses deterministic DE gradients for fast convergence.\"}]",1784201034,23,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"learning-ldpc-codes-with-quantized-density-evolution-over-relaxed-protographs","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/learning-ldpc-codes-with-quantized-density-evolution-over-relaxed-protographs/85090/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What optimization challenge does the document address in LDPC code design?","Question",{"text":75,"@type":76},"Selecting a parity-check matrix or protograph best suited to a prescribed iterative decoder is treated as a difficult combinatorial optimization problem, especially under practical constraints like finite iterations and quantized messages.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why are existing gradient-descent-based code-design methods costly or limited?",{"text":80,"@type":76},"Decoder-in-the-loop strategies require noisy Monte Carlo estimates, line search over soft matrix representations, and often evaluate loss only at integer-valued parity-check matrices, making them expensive for long LDPC codes.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed method optimize relaxed protograph representations?",{"text":84,"@type":76},"Each relaxed protograph entry is interpreted as the probability that the corresponding element equals one, and the loss function is built from DE bit error rate (BER) performance that can be evaluated directly for relaxed protographs. 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