[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84376-en":3,"doc-seo-84376-105":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},84376,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Locality of Curve-Decoding and Improved Proximity Gaps","Proximity gaps describe how an error-correcting code relates to degree-ℓ curves: for any such curve over an alphabet, either all curve points are close to the code or almost all are far. Prior work gave near-optimal proximity gaps for several structured and random code families, but random ensembles suffered worse parameters that deteriorated as the curve degree increased. This work improves proximity gaps for random linear, Reed–Solomon with random evaluation points, and Gallager’s LDPC ensemble, matching subspace-design performance via a black-box transfer technique using a row-span constrained LCL framework and a curve-decoding equivalence theorem.","arXiv :2607 .085 16v 1 [ cs .IT] 9 Jul 2026  \nLocality of Curve-Decoding and Improved Proximity Gaps Rohan Goyal∗ Venkatesan Guruswami † Yihang Sun‡ Mary Wootters§  \nAbstract  \nProximity gaps are a property of error correcting codes that arise in the study of Interactive Oracle Proofs (IOPs) and Succinct Non-interactive Arguments of Zero Knowledge (SNARKs) . Informally, we say that a code C ⊂ Σn exhibits a proximity gap (with respect to degree-ℓ curves) if for any degree-ℓ curve u (x) ∈ Σn , either every point on u (x) is close to C , or else most of them are far from C.  \nRecent work [GG25] has established near-optimal proximity gaps for many families of codes, including subspace design codes, as well as random ensembles like random linear codes, ReedSolomon codes with random evaluation points, and Gallager’s ensemble of LDPC codes. However, the parameters for these latter randomized ensembles are worse than the parameters for subspace design codes, and degrade as the degree ℓ increases.  \nIn this work, we obtain improved proximity gaps for random ensembles of codes, including random linear codes, Reed-Solomon codes with random evaluation points, and Gallager’s ensemble. Quantitatively, our results for these random ensembles match the results that [GG25] attained for subspace design codes. In fact, our techniques are a black-box transference from subspace design codes: Any progress on subspace design codes will automatically lead to analogous progress for these random ensembles.  \nTo obtain our results, we extend the Local Coordinate-wise Linear (LCL) property framework developed in [LMS25 , BCDZ25] to a row-span constrained version. This allows us to cast curvedecodability—a property that implies proximity gaps—directly as an (row-span constrained) LCL property, and make use of that machinery. In contrast, because curve-decodability isnot obviously a (vanilla) LCL property, prior work had worked with a proxy property instead, leading to the aforementioned parameter losses. In addition, we extend the framework to also show an equivalence theorem for Gallager’s ensemble of random LDPC codes and random linear codes for our row-span constrained LCL properties.  \n∗ Massachusetts Institute [of Technology.](of Technology. rohan_g@mit.edu)[ rohan_g@mit.edu](of Technology. rohan_g@mit.edu)[ ](of Technology. rohan_g@mit.edu)†University of California, [Berkeley.](Berkeley. venkatg@berkeley.edu)[ venkatg@berkeley.edu](Berkeley. venkatg@berkeley.edu)[ ](Berkeley. venkatg@berkeley.edu)‡[Stanford University.](Stanford University. kimisun@stanford.edu)[ kimisun@stanford.edu](Stanford University. kimisun@stanford.edu)  \n§ [Stanford University.](Stanford University. marykw@stanford.edu)[ marykw@stanford.edu](Stanford University. marykw@stanford.edu)  \nContents  \n1 Introduction 3  \n1.1 Our Contributions ..................................... 4  \n1.2 Techniques: Curve-Decoding as a Row-Span Constrained LCL Property ....... 5  \n1.3 Related Works ....................................... 7  \n1.4 Discussion and Open Problems .............................. 8  \n2 Preliminaries 9  \n2.1 Proximity Gaps and (Mutual) Correlated Agreement .................. 10  \n2.2 Subspace-Design Codes ................................... 12  \n2.3 Notations and Conventions ................................ 13  \n3 Row-span Constrained Local Properties 13  \n3.1 Generalizing Local Profiles ................................. 13  \n3.2 Behavior Under Quotients ................................. 16  \n3.3 Thresholds for Random Ensembles of Codes ....................... 18  \n3.3.1 Random Linear Codes ............................... 18  \n3.3.2 Random Reed-Solomon Codes ........................... 20  \n3.3.3 Random LDPCs (Gallager’s Ensemble) ...................... 21  \n3.4 Threshold for Subspace Design Codes .......................... 23  \n4 Curve-Decoding as a Row-Span Constrained LCL Property 25  \n5 Improved Proximity Gaps 28  \n5.1 Random Linear Codes .................................","cbCairEmJ4O5RW7O","https://ap.wps.com/l/cbCairEmJ4O5RW7O","pdf",768641,1,35,"English","en",105,"# 1 Introduction\n## 1.1 Our Contributions\n## 1.2 Techniques: Curve-Decoding as a Row-Span Constrained LCL Property\n## 1.3 Related Works\n## 1.4 Discussion and Open Problems\n# 2 Preliminaries\n## 2.1 Proximity Gaps and (Mutual) Correlated Agreement\n## 2.2 Subspace-Design Codes\n## 2.3 Notations and Conventions\n# 3 Row-span Constrained Local Properties\n## 3.1 Generalizing Local Profiles\n## 3.2 Behavior Under Quotients\n## 3.3 Thresholds for Random Ensembles of Codes\n## 3.4 Threshold for Subspace Design Codes\n# 4 Curve-Decoding as a Row-Span Constrained LCL Property\n# 5 Improved Proximity Gaps\n## 5.1 Random Linear Codes\n## 5.2 Random Reed-Solomon Codes\n## 5.3 Random LDPC Codes\n# A Appendix: Gallager’s Ensemble","[{\"question\":\"What is a proximity gap for a code with respect to degree-ℓ curves?\",\"answer\":\"A code has a proximity gap if, for any degree-ℓ curve over the alphabet, either every point on the curve is close to the code or a large fraction of points are far from the code.\"},{\"question\":\"Which code families receive improved proximity-gap parameters in this work?\",\"answer\":\"The results improve proximity gaps for random linear codes, Reed–Solomon codes with random evaluation points, and Gallager’s ensemble of random LDPC codes.\"},{\"question\":\"How does the paper obtain better parameters for random ensembles?\",\"answer\":\"It extends the Local Coordinate-wise Linear (LCL) framework to a row-span constrained version, recasts curve-decodability as a row-span constrained LCL property, and uses a black-box transference from subspace-design codes to random ensembles.\"}]",1784195180,88,{"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},"locality-of-curve-decoding-and-improved-proximity-gaps","",{"@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/locality-of-curve-decoding-and-improved-proximity-gaps/84376/",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 is a proximity gap for a code with respect to degree-ℓ curves?","Question",{"text":75,"@type":76},"A code has a proximity gap if, for any degree-ℓ curve over the alphabet, either every point on the curve is close to the code or a large fraction of points are far from the code.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which code families receive improved proximity-gap parameters in this work?",{"text":80,"@type":76},"The results improve proximity gaps for random linear codes, Reed–Solomon codes with random evaluation points, and Gallager’s ensemble of random LDPC codes.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper obtain better parameters for random ensembles?",{"text":84,"@type":76},"It extends the Local Coordinate-wise Linear (LCL) framework to a row-span constrained version, recasts curve-decodability as a row-span constrained LCL property, and uses a black-box transference from subspace-design codes to random 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