{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/107954"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/107954","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On error correcting codes for distributed storage","abstract":"Two popular directions of error correcting codes for distributed storage are codes with additional recovery or regenerating properties. First we have codes for additional recovery properties. Codewords in array format find applications in disk storage where columns are stored on different disks in combination with parity checks across disks that protect data against disk failures. The addition of global parities protects against sector failures on any of the disks while keeping storage overhead low. We construct sector-disk array codes that tolerate any combination of two disk failures and three sector failures with minimal overhead. This constructs for the first time codes with these parameters without relying on exhaustive search. In the regenerating direction we have some modified layered codes in a two stage construction that gives regenerating codes with small field size. For more general parameters we define a Johnson graph code as a subspace of labelings of the vertices in a Johnson graph with the property that labelings are uniquely determined by their restriction to vertex neighborhoods specified by the parameters of the code. We give a construction and main properties for the codes. We show their role in the concatenation of layered codes to give regenerating codes for storage systems. Focusing on the Minimum Storage regenerating (MSR) point with $d=n-1$, we present graphical representations of codes with parameters \\\\ $((n,k,d), (\\alpha, \\beta)) = ((qt, q(t-1), qt-1),(q^t, q^{t-1}))$ over small field size.","abstract_html":"Two popular directions of error correcting codes for distributed storage are codes with additional recovery or regenerating properties. First we have codes for additional recovery properties. Codewords in array format find applications in disk storage where columns are stored on different disks in combination with parity checks across disks that protect data against disk failures. The addition of global parities protects against sector failures on any of the disks while keeping storage overhead low. We construct sector-disk array codes that tolerate any combination of two disk failures and three sector failures with minimal overhead. This constructs for the first time codes with these parameters without relying on exhaustive search. In the regenerating direction we have some modified layered codes in a two stage construction that gives regenerating codes with small field size. For more general parameters we define a Johnson graph code as a subspace of labelings of the vertices in a Johnson graph with the property that labelings are uniquely determined by their restriction to vertex neighborhoods specified by the parameters of the code. We give a construction and main properties for the codes. We show their role in the concatenation of layered codes to give regenerating codes for storage systems. Focusing on the Minimum Storage regenerating (MSR) point with $d=n-1$, we present graphical representations of codes with parameters \\\\ <span class=\"etd-inline-math\">((n,k,d), (&alpha;, &beta;)) = ((qt, q(t-1), qt-1),(q<sup>t</sup>, q<sup>t-1</sup>))</span> over small field size.","abstract_has_math":true,"creators":["Li, Xiao"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Duursma, Iwan","Reznick, Bruce","Yong, Alexander","Milenkovic, Olgica"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-08-26T21:54:42Z","date_published":"2020-08-26T21:54:42Z","updated_at":"2026-07-22T22:24:47Z","subjects":["Coding Theory, Distributed Storage"],"languages":["en"],"rights":["Copyright 2020 Xiao Li"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/107954","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Duursma, Iwan","Reznick, Bruce","Yong, Alexander","Milenkovic, Olgica"]},{"key":"dc:creator","label":"Author","values":["Li, Xiao"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2020-08-26T21:54:42Z","2020-05-05","2020-05"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Coding Theory, Distributed Storage"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2020 Xiao Li"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/107954"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Two popular directions of error correcting codes for distributed storage are codes with additional recovery or regenerating properties. First we have codes for additional recovery properties. Codewords in array format find applications in disk storage where columns are stored on different disks in combination with parity checks across disks that protect data against disk failures. The addition of global parities protects against sector failures on any of the disks while keeping storage overhead low. We construct sector-disk array codes that tolerate any combination of two disk failures and three sector failures with minimal overhead. This constructs for the first time codes with these parameters without relying on exhaustive search. In the regenerating direction we have some modified layered codes in a two stage construction that gives regenerating codes with small field size. For more general parameters we define a Johnson graph code as a subspace of labelings of the vertices in a Johnson graph with the property that labelings are uniquely determined by their restriction to vertex neighborhoods specified by the parameters of the code. We give a construction and main properties for the codes. We show their role in the concatenation of layered codes to give regenerating codes for storage systems. Focusing on the Minimum Storage regenerating (MSR) point with $d=n-1$, we present graphical representations of codes with parameters \\\\ $((n,k,d), (\\alpha, \\beta)) = ((qt, q(t-1), qt-1),(q^t, q^{t-1}))$ over small field size.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-08-25 without embargo terms","The student, Xiao Li, accepted the attached license on 2020-04-30 at 21:12.","The student, Xiao Li, submitted this Dissertation for approval on 2020-04-30 at 21:17.","This Dissertation was approved for publication on 2020-05-05 at 08:29.","DSpace SAF Submission Ingestion Package generated from Vireo submission #15139 on 2020-08-25 at 17:10:23","Made available in DSpace on 2020-08-26T21:54:42Z (GMT). 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The addition of global parities protects against sector failures on any of the disks while keeping storage overhead low. We construct sector-disk array codes that tolerate any combination of two disk failures and three sector failures with minimal overhead. This constructs for the first time codes with these parameters without relying on exhaustive search. In the regenerating direction we have some modified layered codes in a two stage construction that gives regenerating codes with small field size. For more general parameters we define a Johnson graph code as a subspace of labelings of the vertices in a Johnson graph with the property that labelings are uniquely determined by their restriction to vertex neighborhoods specified by the parameters of the code. We give a construction and main properties for the codes. We show their role in the concatenation of layered codes to give regenerating codes for storage systems. Focusing on the Minimum Storage regenerating (MSR) point with $d=n-1$, we present graphical representations of codes with parameters \\\\ $((n,k,d), (\\alpha, \\beta)) = ((qt, q(t-1), qt-1),(q^t, q^{t-1}))$ over small field size.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-08-25 without embargo terms","The student, Xiao Li, accepted the attached license on 2020-04-30 at 21:12.","The student, Xiao Li, submitted this Dissertation for approval on 2020-04-30 at 21:17.","This Dissertation was approved for publication on 2020-05-05 at 08:29.","DSpace SAF Submission Ingestion Package generated from Vireo submission #15139 on 2020-08-25 at 17:10:23","Made available in DSpace on 2020-08-26T21:54:42Z (GMT). No. of bitstreams: 2 LI-DISSERTATION-2020.pdf: 517839 bytes, checksum: 3712d9afda79d7c086ea70e5ba208199 (MD5) LICENSE.txt: 4204 bytes, checksum: 01cfea356110d2bad0785e127229fdad (MD5) Previous issue date: 2020-05-05"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/107954"],"dc:language":["en"],"dc:rights":["Copyright 2020 Xiao Li"],"dc:subject":["Coding Theory, Distributed Storage"],"dc:title":["On error correcting codes for distributed storage"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:47Z"}