{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86832"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86832","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On Dihedral Codes and the Double Circulant Conjecture for Binary Extended Quadratic Residue Codes","abstract":"Let p be a prime such that p ≡ -1 mod 8. Let k = (p + 1)/2 and write k = 2mq, q odd. Let S = F2[x]/⟨1 + xk⟩ where F2 is the Galois field of two elements. We identify the binary extended quadratic residue codes of length 2k as principal left ideals in the group algebra F2Dk where Dk is the dihedral group of order 2 k. We prove that these codes have a double circulant presentation in the following three cases: (1) q = 1. (2) q is a prime and 2 is a primitive root modulo q . (3) Let X be the class of x in S, and sigma the algebra automorphisms on S that sends Xi to X -i. Factor 1 + xq over F2 into irreducible factors. If the class of those factors in S is fixed by sigma up to a unit, then the codes have a double circulant presentation.","abstract_html":"Let p be a prime such that p ≡ -1 mod 8. Let k = (p + 1)/2 and write k = 2mq, q odd. Let S = F2[x]/⟨1 + xk⟩ where F2 is the Galois field of two elements. We identify the binary extended quadratic residue codes of length 2k as principal left ideals in the group algebra F2Dk where Dk is the dihedral group of order 2 k. We prove that these codes have a double circulant presentation in the following three cases: (1) q = 1. (2) q is a prime and 2 is a primitive root modulo q . (3) Let X be the class of x in S, and sigma the algebra automorphisms on S that sends Xi to X -i. Factor 1 + xq over F2 into irreducible factors. If the class of those factors in S is fixed by sigma up to a unit, then the codes have a double circulant presentation.","abstract_has_math":false,"creators":["Musa, Mona Barakat"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Boston, Nigel"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:19:45Z","date_published":"2015-09-28T15:19:45Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3130989"],"render_values":[{"text":"(MiAaPQ)AAI3130989","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86832","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Boston, Nigel"]},{"key":"dc:creator","label":"Author","values":["Musa, Mona Barakat"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:19:45Z","10000-01-01","2004"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"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":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86832","(MiAaPQ)AAI3130989"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let p be a prime such that p ≡ -1 mod 8. Let k = (p + 1)/2 and write k = 2mq, q odd. Let S = F2[x]/⟨1 + xk⟩ where F2 is the Galois field of two elements. We identify the binary extended quadratic residue codes of length 2k as principal left ideals in the group algebra F2Dk where Dk is the dihedral group of order 2 k. We prove that these codes have a double circulant presentation in the following three cases: (1) q = 1. (2) q is a prime and 2 is a primitive root modulo q . (3) Let X be the class of x in S, and sigma the algebra automorphisms on S that sends Xi to X -i. Factor 1 + xq over F2 into irreducible factors. If the class of those factors in S is fixed by sigma up to a unit, then the codes have a double circulant presentation.","Made available in DSpace on 2015-09-28T15:19:45Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3130989.pdf: 3188162 bytes, checksum: 172d20d68bfa2e40184029df24a1a67d (MD5) Previous issue date: 2004","Embargo set by: Seth Robbins for item 88113 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","72 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2004."]},{"key":"dc:title","label":"Title","values":["On Dihedral Codes and the Double Circulant Conjecture for Binary Extended Quadratic Residue Codes"]}]}],"canonical_facts":{"dc:contributor":["Boston, Nigel"],"dc:creator":["Musa, Mona Barakat"],"dc:date":["2015-09-28T15:19:45Z","10000-01-01","2004"],"dc:description":["Let p be a prime such that p ≡ -1 mod 8. Let k = (p + 1)/2 and write k = 2mq, q odd. Let S = F2[x]/⟨1 + xk⟩ where F2 is the Galois field of two elements. We identify the binary extended quadratic residue codes of length 2k as principal left ideals in the group algebra F2Dk where Dk is the dihedral group of order 2 k. We prove that these codes have a double circulant presentation in the following three cases: (1) q = 1. (2) q is a prime and 2 is a primitive root modulo q . (3) Let X be the class of x in S, and sigma the algebra automorphisms on S that sends Xi to X -i. Factor 1 + xq over F2 into irreducible factors. If the class of those factors in S is fixed by sigma up to a unit, then the codes have a double circulant presentation.","Made available in DSpace on 2015-09-28T15:19:45Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3130989.pdf: 3188162 bytes, checksum: 172d20d68bfa2e40184029df24a1a67d (MD5) Previous issue date: 2004","Embargo set by: Seth Robbins for item 88113 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","72 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2004."],"dc:identifier":["http://hdl.handle.net/2142/86832","(MiAaPQ)AAI3130989"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["On Dihedral Codes and the Double Circulant Conjecture for Binary Extended Quadratic Residue Codes"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:28Z"}