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Divisible Designs, Laguerre Geometry, and Beyond

Hans Havlicek

Abstract

In these notes we aim at bringing together design theory and projective geometry over a ring. Both disciplines are well established, but the results on the interaction between them seem to be rare and scattered over the literature. Thus our main goal is to present the basics from either side, to develop, or at least sketch, the principal connections between them, and to make recommendations for further reading. There is no attempt to provide encyclopedic coverage with expansive notes and references.

Divisible Designs, Laguerre Geometry, and Beyond

Abstract

In these notes we aim at bringing together design theory and projective geometry over a ring. Both disciplines are well established, but the results on the interaction between them seem to be rare and scattered over the literature. Thus our main goal is to present the basics from either side, to develop, or at least sketch, the principal connections between them, and to make recommendations for further reading. There is no attempt to provide encyclopedic coverage with expansive notes and references.

Paper Structure

This paper contains 24 sections, 30 theorems, 126 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Divisible Designs
  3. Basic concepts and first examples
  4. Group actions
  5. A theorem of Spera
  6. Divisible designs and constant weight codes
  7. Notes and further references
  8. Laguerre Geometry
  9. Real Laguerre geometry
  10. The affine and the projective line over a ring
  11. The distant relation
  12. Chain geometries
  13. Local rings, local algebras, and Laguerre algebras
  14. Notes and further references
  15. Divisible Designs via $\mathop{\mathrm{GL}}\nolimits_2$-Actions
  16. ...and 9 more sections

Key Result

Theorem 2.1.8

Let ${\mathcal{D}}$ be a $t$-$(s,k,\lambda_t)$-DD with $t\ge 2$ and let $i$ be an integer such that $1\leq i\leq t$. Then ${\mathcal{D}}$ is also an $i$-$(s,k,\lambda_i)$-DD with

Figures (3)

  • Figure 5: Two non-isomorphic $2$-DDs from an octahedron
  • Figure 6: Two cycles with tangent spears, and a family of parallel spears
  • Figure 7: The distant relation on ${\mathbb P}({\mathbb Z}_6)$

Theorems & Definitions (60)

  • Definition 2.1.3
  • Theorem 2.1.8
  • proof
  • Example 2.1.14
  • Theorem 2.2.3
  • Theorem 2.3.2
  • proof
  • Example 2.3.3
  • Corollary 2.3.5
  • proof
  • ...and 50 more