{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/122263"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/122263","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"A lattice structure on code metrics and beyond f-vectors of matroids","abstract":"Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2025-12-01","abstract_html":"Submission published under a 24 month embargo labeled &#x27;Closed Access&#x27;, the embargo will last until 2025-12-01","abstract_has_math":false,"creators":["Jin, Kexin"],"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","Dodd, Christopher","Mineyev, Igor","Martin, William"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-12","date_published":"2023-12","updated_at":"2026-07-22T22:25:00Z","subjects":["Coding Theory","Algebraic Combinatorics"],"languages":["en","eng"],"rights":["Copyright 2023 Kexin Jin"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/122263","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Duursma, Iwan","Dodd, Christopher","Mineyev, Igor","Martin, William"]},{"key":"dc:creator","label":"Author","values":["Jin, Kexin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2023-12","2023-12-04"]},{"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":["Coding Theory","Algebraic Combinatorics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2023 Kexin Jin"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/122263"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2025-12-01","The student, Kexin Jin, accepted the attached license on 2023-12-01 at 15:10.","The student, Kexin Jin, submitted this Dissertation for approval on 2023-12-01 at 15:15.","This Dissertation was approved for publication on 2023-12-04 at 17:23.","DSpace SAF Submission Ingestion Package generated from Vireo submission #20096 on 2024-03-01 at 13:56:58","Let \\((X, \\leq)\\) be a regular semilattice. Delsarte showed the top fiber of \\(X\\) carries a structure of association schemes. In this thesis, we construct another lattice structure, which allows us to give a unified proof and construction of MacWilliams transforms for different code metrics including the Hamming, the Niederreiter-Rosenbloom-Tsfasman, and the rank metric. Furthermore, the lattice structure can also be connected to association schemes, and the connection allows applications to objects such as the symmetric groups \\(S_n\\). Let \\(M = (E, \\mathcal{I})\\) be a matroid with \\(| E | = n\\) and rank \\(r\\). Let \\(1 \\leq k < r\\), then Mason's ultra log-concavity conjecture states that \\(I_{k}^2 \\geq \\big(1 + \\frac{1}{k}\\big) \\big(1 + \\frac{1}{n-k}\\big) I_{k-1} I_{k+1}\\), where \\(I_{k}\\) denotes the number of independent sets of size \\(k\\) in \\(M\\). In this thesis, we prove an improvement and a generalization of the ultra log-concavity property. We further give a sufficient condition that allows us to tell the ultra log-concavity property holds for constant nullity sets. Furthermore, we give a probabilistic interpretation of the ultra log-concavity property and show a counterexample of synchronicity of two sequences related to matroids."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["A lattice structure on code metrics and beyond f-vectors of matroids"]}]}],"canonical_facts":{"dc:contributor":["Duursma, Iwan","Dodd, Christopher","Mineyev, Igor","Martin, William"],"dc:creator":["Jin, Kexin"],"dc:date":["2023-12","2023-12-04"],"dc:description":["Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2025-12-01","The student, Kexin Jin, accepted the attached license on 2023-12-01 at 15:10.","The student, Kexin Jin, submitted this Dissertation for approval on 2023-12-01 at 15:15.","This Dissertation was approved for publication on 2023-12-04 at 17:23.","DSpace SAF Submission Ingestion Package generated from Vireo submission #20096 on 2024-03-01 at 13:56:58","Let \\((X, \\leq)\\) be a regular semilattice. Delsarte showed the top fiber of \\(X\\) carries a structure of association schemes. In this thesis, we construct another lattice structure, which allows us to give a unified proof and construction of MacWilliams transforms for different code metrics including the Hamming, the Niederreiter-Rosenbloom-Tsfasman, and the rank metric. Furthermore, the lattice structure can also be connected to association schemes, and the connection allows applications to objects such as the symmetric groups \\(S_n\\). Let \\(M = (E, \\mathcal{I})\\) be a matroid with \\(| E | = n\\) and rank \\(r\\). Let \\(1 \\leq k < r\\), then Mason's ultra log-concavity conjecture states that \\(I_{k}^2 \\geq \\big(1 + \\frac{1}{k}\\big) \\big(1 + \\frac{1}{n-k}\\big) I_{k-1} I_{k+1}\\), where \\(I_{k}\\) denotes the number of independent sets of size \\(k\\) in \\(M\\). In this thesis, we prove an improvement and a generalization of the ultra log-concavity property. We further give a sufficient condition that allows us to tell the ultra log-concavity property holds for constant nullity sets. Furthermore, we give a probabilistic interpretation of the ultra log-concavity property and show a counterexample of synchronicity of two sequences related to matroids."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/122263"],"dc:language":["en","eng"],"dc:rights":["Copyright 2023 Kexin Jin"],"dc:subject":["Coding Theory","Algebraic Combinatorics"],"dc:title":["A lattice structure on code metrics and beyond f-vectors of matroids"],"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":"2026-07-22T22:25:00Z"}