2026-09-21T12:55:23Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/122263 2024-03-02 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Duursma, Iwan Dodd, Christopher Mineyev, Igor Martin, William 2023-12 application/pdf en text 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 A lattice structure on code metrics and beyond f-vectors of matroids Jin, Kexin Coding Theory Algebraic Combinatorics 2023-12-04 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. Text eng https://hdl.handle.net/2142/122263 Copyright 2023 Kexin Jin Ph.D. Dissertation Mathematics University of Illinois at Urbana-Champaign Mathematics