[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81885-en":3,"doc-seo-81885-105":31,"detail-sidebar-cat-0-en-105":97},{"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":28,"seo_description":14,"update_tm":29,"read_time":30},81885,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",8,"Research & Report","Function Correcting Codes for Sum Rank Metric","Function-Correcting Codes (FCCs) protect the reliable evaluation of a specified function of a message under channel errors, using less redundancy than traditional error-correcting codes. The paper investigates FCCs in the sum-rank metric, a framework generalizing both the Hamming and rank metrics, and derives general upper and lower bounds on optimal redundancy. A Plotkin-like bound is established for irregular-distance codes and simplified for linear functions, and explicit constructions are provided for function-correcting sum-rank metric codes achieving optimal redundancy.","Function-Correcting Codes for Sum-Rank Metric  \nSanthi Kumari Kammila and B. Sundar Rajan, Life Fellow IEEE  \nDepartment of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India  \n{santhik,[bsrajan}@iisc.ac.in](bsrajan}@iisc.ac.in)  \narXiv :2607 .03857v2 [ cs .IT] 10 Jul 2026  \nAbstract—Function-Correcting Codes (FCCs) are a class of codes designed to protect the evaluation of a specific function of a message against channel errors at a higher level than the level of protection for the message, while requiring significantly less redundancy than conventional error-correcting codes. In this paper, we study function-correcting codes under the sum-rank metric, which is a natural generalization of both the Hamming metric and the rank-metric and also we derive general upper and lower bounds on the optimal redundancy of FCCs in the sumrank metric. In particular, we establish a Plotkin-like bound for irregular-distance codes in sum-rank metric and simplify it for linear functions. Furthermore, we present explicit construction of function-correcting sum-rank metric codes (FCSRCs) for locally binary functions and sum-rank weight functions with optimal redundancy.  \nIndex Terms—function-correcting codes, optimal redundancy, Plotkin-like bound, rank-metric, sum-rank metric, .  \nI. INTRODUCTION  \nThe coding framework for function-correcting codes (FCC) is introduced in [1] to protect a specific attribute (function) of the message that is of interest to the receiver along with the entire message with reduced redundancy. When the function to be protected is a bijective mapping, the FCC coincides with classical error-correcting code (ECC) . In [1] FCCs have been studied for the Hamming metric. We study FCCs for channels matched to the sum-rank metric calling them FunctionCorrecting Sum-Rank Codes (FCSRCs). These are particularly beneficial in multi-shot network coding and distributed storage systems where classical sum-rank metric codes are utilized, as FCSRCs enable reliable recovery of specific function of interest with reduced redundancy compared to classical sumrank metric codes. Rank-metric codes and sum-rank metric codes are reviewed in the following subsection and a brief review of the state-of-art works on FCCs in various settings is given in the subsequent subsection.  \nA. Sum-rank metric codes  \nIn [2] and [3] the authors introduce rank-metric codes asa natural and powerful framework for error control in random linear network coding. Network transmission errors are modeled as additive matrix perturbations and it is shown that the rank of the error matrix captures the effect of adversarial errors in network-coded systems. By lifting rank-metric codes to subspace codes, a direct connection between rank distance and subspace distnace is established providing a rigorous justification for using rank-metric codes to achieve optimal error correction performance in random network coding. Gabidulin introduced a fundamental class of linear codes called Max  \nimum Rank Distance (MRD) codes defined over extension  \nfields that achieve the maximum possible distance under the rank-metric, now known as Gabidulin codes [4] . These are analogous to maximum distance separable (MDS) codes in the Hamming metric and attain the Singleton bound for the rank-metric and are therefore optimal. In [4] an algebraic construction of such codes based on linearized polynomials is presented and efficient decoding algorithms for correcting rank errors are developed. This work is further extended to symmetric rank-metric codes in [5] in which the authors showed that for extension fields of characteristic 2, the field can be represented by symmetric matrices, which leads to the construction of MRD codes consisting entirely of symmetric matrices. These codes achieve maximum rank distance and their vector representations correspond to linear MRD Codes.  \nCodes over sum-rank metric are collection of vectors of matrices and the metric is obta","cbCaisrWc68qPRp2","https://ap.wps.com/l/cbCaisrWc68qPRp2","pdf",293730,7,1,9,"English","en",105,"# I. Introduction\n## A. Sum-rank metric codes\n## B. Function-Correcting Codes (FCC): State-of-art","[{\"question\":\"What is the main goal of Function-Correcting Codes (FCCs)?\",\"answer\":\"FCCs are designed to safeguard the correct evaluation of a chosen function of the transmitted message under channel errors, while using substantially less redundancy than conventional error-correcting codes.\"},{\"question\":\"Why is the sum-rank metric important for this work?\",\"answer\":\"The paper studies FCCs under the sum-rank metric, which naturally generalizes the Hamming metric and the rank metric, enabling analysis and bounds for codes operating across multiple blocks.\"},{\"question\":\"What key theoretical results does the paper present?\",\"answer\":\"It derives general upper and lower bounds on optimal redundancy in the sum-rank setting, establishes a Plotkin-like bound for irregular-distance codes, and simplifies it for linear functions.\"},{\"question\":\"What constructions are provided in the paper?\",\"answer\":\"The work presents explicit constructions of function-correcting sum-rank metric codes for locally binary functions and for sum-rank weight functions, with optimal redundancy.\"}]","Function Correcting Codes for Sum Rank Metric | PDF",1784176863,23,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":92,"head_meta":94,"extra_data":96,"updated_unix":29},"function-correcting-codes-for-sum-rank-metric","",{"@graph":37,"@context":91},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/function-correcting-codes-for-sum-rank-metric/81885/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-08-04","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83,87],{"name":74,"@type":75,"acceptedAnswer":76},"What is the main goal of Function-Correcting Codes (FCCs)?","Question",{"text":77,"@type":78},"FCCs are designed to safeguard the correct evaluation of a chosen function of the transmitted message under channel errors, while using substantially less redundancy than conventional error-correcting codes.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"Why is the sum-rank metric important for this work?",{"text":82,"@type":78},"The paper studies FCCs under the sum-rank metric, which naturally generalizes the Hamming metric and the rank metric, enabling analysis and bounds for codes operating across multiple blocks.",{"name":84,"@type":75,"acceptedAnswer":85},"What key theoretical results does the paper present?",{"text":86,"@type":78},"It derives general upper and lower bounds on optimal redundancy in the sum-rank setting, establishes a Plotkin-like bound for irregular-distance codes, and simplifies it for linear functions.",{"name":88,"@type":75,"acceptedAnswer":89},"What constructions are provided in the paper?",{"text":90,"@type":78},"The work presents explicit constructions of function-correcting sum-rank metric codes for locally binary functions and for sum-rank weight functions, with optimal redundancy.","https://schema.org",{"og:url":53,"og:type":93,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":95,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":98},[99,103,107,111,116,121,125,128,132,135,139],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Story & 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