Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Constructions of $t$-designs from weighing matrices and walk-regular graphs

Gary Greaves, Sho Suda

Abstract

We provide a method to construct $t$-designs from weighing matrices and walk-regular graphs. One instance of our method can produce a $3$-design from any (symmetric or skew-symmetric) conference matrix, thereby providing a partial answer to a question of Gunderson and Semeraro JCTB 2017. We explore variations of our method on some matrices that satisfy certain combinatorial restrictions. In particular, we show that there exist various infinite families of partially balanced incomplete block designs with block size four on the binary Hamming schemes and the $3$-class association schemes attached to symmetric designs, and regular pairwise balanced designs with block sizes three and four.

Constructions of $t$-designs from weighing matrices and walk-regular graphs

Abstract

We provide a method to construct -designs from weighing matrices and walk-regular graphs. One instance of our method can produce a -design from any (symmetric or skew-symmetric) conference matrix, thereby providing a partial answer to a question of Gunderson and Semeraro JCTB 2017. We explore variations of our method on some matrices that satisfy certain combinatorial restrictions. In particular, we show that there exist various infinite families of partially balanced incomplete block designs with block size four on the binary Hamming schemes and the -class association schemes attached to symmetric designs, and regular pairwise balanced designs with block sizes three and four.
Paper Structure (25 sections, 20 theorems, 71 equations, 1 figure, 13 tables)

This paper contains 25 sections, 20 theorems, 71 equations, 1 figure, 13 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Main results and their consequences
  4. Organisation
  5. Notation
  6. Preliminaries
  7. Designs constructed from square matrices
  8. Hypergraphs from square matrices
  9. 3-designs from conference matrices
  10. 2-designs from equiangular tight frames and Hadamard matrices
  11. 1-designs from walk-regular graphs
  12. Partially balanced incomplete block designs
  13. Symmetric association schemes
  14. Skew-Bush type Hadamard matrices and group divisible designs
  15. Extending to three distinct $k \times k$ principal minors
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $A$ be a $v \times v$ complex matrix that has precisely two principal $k \times k$ minors $a$ and $b$. Suppose, for each $i \in \{0,1,\dots,t\}$, the coefficient of $x^{v-k-i}$ of the characteristic polynomial of each principal $(v-i) \times (v-i)$ submatrix of $A$ is constant. Then the hypergra

Figures (1)

  • Figure 1: The matrix $H$ from Section \ref{['sec:sbh']}.

Theorems & Definitions (48)

  • Theorem 1.1: Synopsis of Theorem \ref{['thm:des']}
  • Lemma 1.3: Jacobi
  • Lemma 1.4
  • proof
  • Corollary 1.5
  • Lemma 1.6
  • Corollary 1.7
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • ...and 38 more