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Combinatorics of Complex Maximal Determinant Matrices

Guillermo Nuñez Ponasso

Abstract

This doctoral thesis covers several topics related to the construction and study of maximal determinant matrices with complex entries. The first three chapters are devoted to number-theoretic tools to prove the non-solvability of Gram matrix equations over certain fields, with a focus on combinatorial applications. Chapter 4 gives a survey on Butson-type Hadamard matrices, and shows an improved lower bound on primes $p$ for the existence of $BH(12p, p)$ matrices. Chapter 5 contains the main contributions of the thesis, where the maximal determinant problem for matrices over the m-th roots of unity is discussed, and where new upper and lower bounds, as well as constructions at small orders, are given. Chapter 6 studies maximal determinant matrices over association schemes. Chapter 7 gives an application of design theory to privacy in communications, and it is connected to the rest of the thesis by the use of the theory of quadratic forms.

Combinatorics of Complex Maximal Determinant Matrices

Abstract

This doctoral thesis covers several topics related to the construction and study of maximal determinant matrices with complex entries. The first three chapters are devoted to number-theoretic tools to prove the non-solvability of Gram matrix equations over certain fields, with a focus on combinatorial applications. Chapter 4 gives a survey on Butson-type Hadamard matrices, and shows an improved lower bound on primes for the existence of matrices. Chapter 5 contains the main contributions of the thesis, where the maximal determinant problem for matrices over the m-th roots of unity is discussed, and where new upper and lower bounds, as well as constructions at small orders, are given. Chapter 6 studies maximal determinant matrices over association schemes. Chapter 7 gives an application of design theory to privacy in communications, and it is connected to the rest of the thesis by the use of the theory of quadratic forms.
Paper Structure (68 sections, 283 theorems, 931 equations, 9 figures, 12 tables)

This paper contains 68 sections, 283 theorems, 931 equations, 9 figures, 12 tables.

Table of Contents

  1. Non-solvability of Gram equations
  2. Positive-definite matrices and matrix analysis
  3. Quadratic forms
  4. Witt's Lemma
  5. Hilbert symbols
  6. Invariants of quadratic forms
  7. Invariants of Quadratic forms in Design Theory
  8. Design theory
  9. The Bruck-Ryser-Chowla Theorem
  10. The Bose-Connor Theorem
  11. Non-feasibility tables for parameters of GDDs
  12. An application to maximal determinant matrices
  13. Hermitian Forms and Determinant Obstructions
  14. Hermitian forms
  15. Splitting of prime ideals
  16. ...and 53 more sections

Key Result

Theorem 1

Suppose that there is a symmetric $2$-$(v,k,\lambda)$ design. Then,

Figures (9)

  • Figure 1: The Fano plane.
  • Figure 2: Visualisation of a UPIR system
  • Figure 3: The smallest non-trivial generalised quadrangle.
  • Figure 4: A $\mathop{\mathrm{BH}}\nolimits(12,3)$ matrix.
  • Figure 5: A $\mathop{\mathrm{BH}}\nolimits(24,3)$ matrix.
  • ...and 4 more figures

Theorems & Definitions (605)

  • Theorem : Bruck-Ryser-Chowla
  • Theorem
  • Theorem
  • Theorem
  • Proposition
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • ...and 595 more