[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83845-en":3,"doc-seo-83845-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83845,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Minimum Distances of LDPC Codes in 5G Standard","The study develops multiple approaches to bound the minimum distances of quasi-cyclic LDPC codes used in the 5G NR standard. It establishes that high-rate BG1 5GLDPC codes [9984, 8448] and low-rate codes [25344, 8448] have minimum distances in the ranges {8,...,14} and {22,...,57}. The work also introduces a new early termination method using circulant modular reduction that substantially reduces LDPC decoder syndrome-calculation complexity.","arXiv :2607 .047 16v 1 [ cs .IT] 6 Jul 2026  \nMINIMUM DISTANCES OF LDPC CODES IN 5G STANDARD  \nV.R. DANILKO, I.YU. MOGILNYKH, AND YA.A. TIKHOMOLOV  \nAbstract: We propose several approaches for bounding the minimum distances of the family of quasi-cyclic LDPC codes in the 5G NR standard. In particular, we show that the high-rate [9984 , 8448] and the low-rate [25344 , 8448] BG1 5GLDPC codes have minimum distances in the ranges {8,..., 14} and {22,..., 57}, respectively. Also we propose a new early termination approach based on circulant modular reduction, which significantly lowers syndrome calculation complexity for the LDPC decoder.  \n1 Introduction  \nQuasi-cyclic LDPC codes used in the 5G NR standard [1] form a distinctive class of errorcorrecting codes that combine strong performance with efficient hardware implementation. In this paper, we consider the minimum distance problem for 5G LDPC codes, which is known to be NP-hard for the general class of linear codes [24] . Special attention is given to the codes formed by the first 4 and 6 parity circulant blocks, referred to hereafter asthe 4-layer and 6-layer codes, respectively, for the following reasons.  \nThe full 5G LDPC parity-check matrix [18] has an unusual structure for a quasi-cyclic LDPC code: it is obtained from four parity layers by appending a sparse matrix below them and a large identity matrix in the lower-right corner, see Fig. 3. On one hand, this structure makes encoding very computationally efficient [16] . On the other hand, one may expect that the presence of weight one columns in the parity-check matrix degrades the minimum distance of the code and, consequently, its decoding performance. However, due to the careful choice of circulant blocks in the matrix construction [18] and the retransmission mechanism of the 5G communication system, this degradation is significantly reduced when a low-complexity approximate decoder such as the layered min-sum decoder is used, although it remains visible for high-rate codes; see Fig. 1.Moreover, the decoder is prone to undetected errors, see Fig. 9. This phenomenon can be explained by the short minimum distances of high-rate 5G LDPC codes, which we establish in this study.  \nThe 4-layer code also plays a central role in the 5G HARQ retransmission scheme. In this scheme, although only certain parts of a codeword are transmitted in each retransmission, the initial transmission (redundancy version 0, RV0) starts at the third circulant block and includes all the remaining information bits and partially the leftmost parity bits. This fact makes RV0 closely related to the 4-layer code.  \nThe [9984 , 8448] code, which is the 6-layer code with maximum circulant size 384, was also adopted as part of the SDA-OCT communication standard [19] . In this standard, the error-detection properties are strengthened. Each information block contains a 144-bit header followed by a 16-bit CRC, and the block is additionally protected by 800 parity bits of a convolutional code [19, Section [3.4.5.2](3.4.5.2)] . As a result, the information-block errordetection capabilities of this standard are much stronger than those in 5G, partly to cope with the use of a simpler retransmission ARQ protocol in SDA-OCT.  \nV. R. Danilko and Ya. A. Tikhomolov are with Novosibirsk State University, Novosibirsk, Russia and I. Yu. Mogilnykh is with the Sobolev Institute of Mathematics, Novosibirsk, Russia. The work was performed according to the Government research assignment for IM SB RAS, Project No. FWNF-2026-0011 .  \n2 V.R. DANILKO, I.YU. MOGILNYKH, AND YA.A. TIKHOMOLOV  \nThe next section presents the necessary preliminaries, including basic definitions, notations, and a description of 5G LDPC codes.  \nThe main goal ofthis work is to derive upper and lower bounds on the minimum distances of the 5G LDPC codes. To this end, in Section 3 we exploit the block structure of 5G LDPC parity-check matrices to adapt the classical Vontobel–Smarandache construction [20]","cbCaicmnvK8ROUHa","https://ap.wps.com/l/cbCaicmnvK8ROUHa","pdf",975335,1,24,"English","en",105,"# Introduction\n## Minimum distance bounds\n## Decoder early termination and complexity","[{\"question\":\"What problem does the paper address for 5G NR LDPC codes?\",\"answer\":\"It derives upper and lower bounds on the minimum distances of quasi-cyclic LDPC codes defined in the 5G NR standard.\"},{\"question\":\"What minimum-distance ranges are reported for the BG1 5GLDPC codes?\",\"answer\":\"For the high-rate [9984, 8448] code the minimum distance lies in {8,...,14}, and for the low-rate [25344, 8448] code it lies in {22,...,57}.\"},{\"question\":\"How does the proposed early termination approach reduce decoder complexity?\",\"answer\":\"It uses circulant modular reduction to lower syndrome calculation complexity in the LDPC decoder, enabling earlier stopping conditions in practice.\"}]",1784190936,60,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"minimum-distances-of-ldpc-codes-in-5g-standard","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/minimum-distances-of-ldpc-codes-in-5g-standard/83845/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 problem does the paper address for 5G NR LDPC codes?","Question",{"text":75,"@type":76},"It derives upper and lower bounds on the minimum distances of quasi-cyclic LDPC codes defined in the 5G NR standard.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What minimum-distance ranges are reported for the BG1 5GLDPC codes?",{"text":80,"@type":76},"For the high-rate [9984, 8448] code the minimum distance lies in {8,...,14}, and for the low-rate [25344, 8448] code it lies in {22,...,57}.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed early termination approach reduce decoder complexity?",{"text":84,"@type":76},"It uses circulant modular reduction to lower syndrome calculation complexity in the LDPC decoder, enabling earlier stopping conditions in practice.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":28,"slug":108},5,"Comic","comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]