A lattice structure on code metrics and beyond f-vectors of matroids

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_k² \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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Jin, Kexin

Contributors dc:contributor

• Duursma, Iwan
• Dodd, Christopher
• Mineyev, Igor
• Martin, William

• Coding Theory
• Algebraic Combinatorics

Statement dc:rights

• Copyright 2023 Kexin Jin

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/122263

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F122263