{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/37929"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/37929","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Local list decoding of homomorphisms","abstract":"We investigate the local-list decodability of codes whose codewords are group homomorphisms. The study of such codes was intiated by Goldreich and Levin with the seminal work on decoding the Hadamard code. Many of the recent abstractions of their initial algorithm focus on Locally Decodable Codes (LDC's) over finite fields. We derive our algorithmic approach from the list decoding of the Reed-Muller code over finite fields proposed by Sudan, Trevisan and Vadhan. Given an abelian group G and a fixed abelian group H, we give combinatorial bounds on the number of homomorphisms that have agreement 6 with an oracle-access function f : G --> H. Our bounds are polynomial in , where the degree of the polynomial depends on H. Also, depends on the distance parameter of the code, namely we consider to be slightly greater than 1-minimum distance. Furthermore, we give a local-list decoding algorithm for the homomorphisms that agree on a 3 fraction of the domain with a function f, the running time of which is poly(1/e, log G).","abstract_html":"We investigate the local-list decodability of codes whose codewords are group homomorphisms. The study of such codes was intiated by Goldreich and Levin with the seminal work on decoding the Hadamard code. Many of the recent abstractions of their initial algorithm focus on Locally Decodable Codes (LDC's) over finite fields. We derive our algorithmic approach from the list decoding of the Reed-Muller code over finite fields proposed by Sudan, Trevisan and Vadhan. Given an abelian group G and a fixed abelian group H, we give combinatorial bounds on the number of homomorphisms that have agreement 6 with an oracle-access function f : G --> H. Our bounds are polynomial in , where the degree of the polynomial depends on H. Also, depends on the distance parameter of the code, namely we consider to be slightly greater than 1-minimum distance. Furthermore, we give a local-list decoding algorithm for the homomorphisms that agree on a 3 fraction of the domain with a function f, the running time of which is poly(1/e, log G).","abstract_has_math":false,"creators":["Grigorescu, Elena, Ph. D. Massachusetts Institute of Technology"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.","school":null,"contributors":[],"advisors":["Madhu Sudan."],"committee_chairs":[],"committee_members":[],"year":2006,"date_issued":"2006","date_published":"2006","updated_at":"2026-07-22T22:21:53Z","subjects":["Electrical Engineering and Computer Science."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/37929","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Madhu Sudan."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science."]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science."]},{"key":"dc:creator","label":"Author","values":["Grigorescu, Elena, Ph. D. 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They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1721.1/37929"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.","Includes bibliographical references (leaves 47-49)."]},{"key":"dc:description.abstract","label":"Abstract","values":["We investigate the local-list decodability of codes whose codewords are group homomorphisms. The study of such codes was intiated by Goldreich and Levin with the seminal work on decoding the Hadamard code. Many of the recent abstractions of their initial algorithm focus on Locally Decodable Codes (LDC's) over finite fields. We derive our algorithmic approach from the list decoding of the Reed-Muller code over finite fields proposed by Sudan, Trevisan and Vadhan. Given an abelian group G and a fixed abelian group H, we give combinatorial bounds on the number of homomorphisms that have agreement 6 with an oracle-access function f : G --> H. Our bounds are polynomial in , where the degree of the polynomial depends on H. Also, depends on the distance parameter of the code, namely we consider to be slightly greater than 1-minimum distance. Furthermore, we give a local-list decoding algorithm for the homomorphisms that agree on a 3 fraction of the domain with a function f, the running time of which is poly(1/e, log G)."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["S.M."]},{"key":"dc:title","label":"Title","values":["Local list decoding of homomorphisms"]}]}],"canonical_facts":{"dc:contributor.advisor":["Madhu Sudan."],"dc:contributor.department":["Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science."],"dc:contributor.other":["Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science."],"dc:creator":["Grigorescu, Elena, Ph. D. 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Given an abelian group G and a fixed abelian group H, we give combinatorial bounds on the number of homomorphisms that have agreement 6 with an oracle-access function f : G --> H. Our bounds are polynomial in , where the degree of the polynomial depends on H. Also, depends on the distance parameter of the code, namely we consider to be slightly greater than 1-minimum distance. Furthermore, we give a local-list decoding algorithm for the homomorphisms that agree on a 3 fraction of the domain with a function f, the running time of which is poly(1/e, log G)."],"dc:description.degree":["S.M."],"dc:identifier.uri":["http://hdl.handle.net/1721.1/37929"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Electrical Engineering and Computer Science."],"dc:title":["Local list decoding of homomorphisms"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:21:53Z"}