{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/98818"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/98818","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Repairing Cartesian Codes with Linear Exact Repair Schemes","abstract":"In this paper, we develop a scheme to recover a single erasure when using a Cartesian code,in the context of a distributed storage system. Particularly, we develop a scheme withconsiderations to minimize the associated bandwidth and maximize the associateddimension. The problem of recovering a missing node's data exactly in a distributedstorage system is known as theexact repair problem. Previous research has studied theexact repair problem for Reed-Solomon codes. 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We focus on Cartesian codes, and show wecan enact the recovery using a linear exact repair scheme framework, similar to the oneoutlined by Guruswami and Wooters in 2017.","abstract_has_math":false,"creators":["Valvo, Daniel William"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Matthews, Gretchen L."],"committee_members":["Mihalcea, Constantin Leonardo","Orr, Daniel D."],"year":2020,"date_issued":"2020-06-10","date_published":"2020-06-10","updated_at":"2026-07-22T22:19:59Z","subjects":["Cartesian code","Reed-Solomon code","exact repair schemes","finite fields","distributed storage networks","multivariate polynomials"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:26553"],"render_values":[{"text":"vt_gsexam:26553","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/98818","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Matthews, Gretchen L."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Mihalcea, Constantin Leonardo","Orr, Daniel D."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Valvo, Daniel William"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2020-06-11T08:00:24Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2020-06-11T08:00:24Z"]},{"key":"dc:date.issued","label":"Date","values":["2020-06-10"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Cartesian code","Reed-Solomon code","exact repair schemes","finite fields","distributed storage networks","multivariate polynomials"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:26553"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/98818"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this paper, we develop a scheme to recover a single erasure when using a Cartesian code,in the context of a distributed storage system. Particularly, we develop a scheme withconsiderations to minimize the associated bandwidth and maximize the associateddimension. The problem of recovering a missing node's data exactly in a distributedstorage system is known as theexact repair problem. Previous research has studied theexact repair problem for Reed-Solomon codes. We focus on Cartesian codes, and show wecan enact the recovery using a linear exact repair scheme framework, similar to the oneoutlined by Guruswami and Wooters in 2017."]},{"key":"dc:description.abstractgeneral","label":"General Abstract","values":["Distributed storage systems are systems which store a single data file over multiple storage nodes. Each storage node has a certain storage efficiency, the \"space\" required to store the information on that node. The value of these systems, is their ability to safely store data for extended periods of time. We want to design distributed storage systems such that if one storage node fails, we can recover it from the data in the remaining nodes. Recovering a node from the data stored in the other nodes requires the nodes to communicate data with each other. Ideally, these systems are designed to minimize the bandwidth, the inter-nodal communication required to recover a lost node, as well as maximize the storage efficiency of each node. A great mathematical framework to build these distributed storage systems on is erasure codes. In this paper, we will specifically develop distributed storage systems that use Cartesian codes. We will show that in the right setting, these systems can have a very similar bandwidth to systems build from Reed-Solomon codes, without much loss in storage efficiency."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["Repairing Cartesian Codes with Linear Exact Repair Schemes"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Matthews, Gretchen L."],"dc:contributor.committeemember":["Mihalcea, Constantin Leonardo","Orr, Daniel D."],"dc:contributor.department":["Mathematics"],"dc:creator":["Valvo, Daniel William"],"dc:date.accessioned":["2020-06-11T08:00:24Z"],"dc:date.available":["2020-06-11T08:00:24Z"],"dc:date.issued":["2020-06-10"],"dc:description.abstract":["In this paper, we develop a scheme to recover a single erasure when using a Cartesian code,in the context of a distributed storage system. Particularly, we develop a scheme withconsiderations to minimize the associated bandwidth and maximize the associateddimension. The problem of recovering a missing node's data exactly in a distributedstorage system is known as theexact repair problem. Previous research has studied theexact repair problem for Reed-Solomon codes. We focus on Cartesian codes, and show wecan enact the recovery using a linear exact repair scheme framework, similar to the oneoutlined by Guruswami and Wooters in 2017."],"dc:description.abstractgeneral":["Distributed storage systems are systems which store a single data file over multiple storage nodes. Each storage node has a certain storage efficiency, the \"space\" required to store the information on that node. The value of these systems, is their ability to safely store data for extended periods of time. We want to design distributed storage systems such that if one storage node fails, we can recover it from the data in the remaining nodes. Recovering a node from the data stored in the other nodes requires the nodes to communicate data with each other. Ideally, these systems are designed to minimize the bandwidth, the inter-nodal communication required to recover a lost node, as well as maximize the storage efficiency of each node. A great mathematical framework to build these distributed storage systems on is erasure codes. In this paper, we will specifically develop distributed storage systems that use Cartesian codes. We will show that in the right setting, these systems can have a very similar bandwidth to systems build from Reed-Solomon codes, without much loss in storage efficiency."],"dc:description.degree":["Master of Science"],"dc:format.medium":["ETD"],"dc:identifier.other":["vt_gsexam:26553"],"dc:identifier.uri":["http://hdl.handle.net/10919/98818"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Cartesian code","Reed-Solomon code","exact repair schemes","finite fields","distributed storage networks","multivariate polynomials"],"dc:title":["Repairing Cartesian Codes with Linear Exact Repair Schemes"],"dc:type":["Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:59Z"}