University of Illinois Urbana-Champaign

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

Jin, Kexin

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

Description

Title
A lattice structure on code metrics and beyond f-vectors of matroids
Author(s)
Jin, Kexin
Issue Date
2023-12-04
Director of Research (if dissertation) or Advisor (if thesis)
Duursma, Iwan
Doctoral Committee Chair(s)
Dodd, Christopher
Committee Member(s)
  • Mineyev, Igor
  • Martin, William
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
  • Coding theory
  • algebraic combinatorics
Abstract
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.
Graduation Semester
2023-12
Type of Resource
Thesis
Copyright and License Information
Copyright 2023 Kexin Jin

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