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Some Constructions of Divisible Designs from Laguerre Geometries

Sabine Giese, Hans Havlicek, Ralph-Hardo Schulz

TL;DR

This work gets series of divisible designs from finite Laguerre geometries whose construction was based on a conic in a plane of a three-dimensional projective space and shows a close connection between some of thesedivisible designs.

Abstract

In the nineties, A.G. Spera introduced a construction principle for divisible designs. Using this method, we get series of divisible designs from finite Laguerre geometries. We show a close connection between some of these divisible designs and divisible designs whose construction was based on a conic in a plane of a 3-dimensional projective space.

Some Constructions of Divisible Designs from Laguerre Geometries

TL;DR

This work gets series of divisible designs from finite Laguerre geometries whose construction was based on a conic in a plane of a three-dimensional projective space and shows a close connection between some of thesedivisible designs.

Abstract

In the nineties, A.G. Spera introduced a construction principle for divisible designs. Using this method, we get series of divisible designs from finite Laguerre geometries. We show a close connection between some of these divisible designs and divisible designs whose construction was based on a conic in a plane of a 3-dimensional projective space.

Paper Structure

This paper contains 8 sections, 8 theorems, 4 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary terms, definitions and results
  3. About Divisible Designs
  4. SPERA's construction principle
  5. A Laguerre Geometry over the finite field $\mathrm{GF}(q)$
  6. Construction of divisible designs from a La-guerre geometry
  7. Proof of Theorem \ref{['meinDD']}:
  8. Comparing both constructions

Key Result

Proposition 2.5

Let $\Lambda=(G,X,R)$ be a finite $t-R$-transitive $R$-group, and let $B$ be an $R$-transversal subset of $X$ with $t \le k:=|B|<v:=|X|$, then the incidence structure $D(\Lambda, B)=(X,B^G,S) \hbox{for} B^G=\{B^g \,|\, g\in G\}$ is a $t-(s,k,\lambda_t)$-divisible design with $s=|[x]|$ for some $x \i where $G_B$ denotes the setwise stabiliser of $B$ and $b$ the number of blocks of $D(\Lambda, B)$.

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2: Definition of divisible designs
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5: A.G. Spera, sp1
  • Remark 2.6
  • Theorem 2.7: Cerroni, Schulz
  • Definition 2.8
  • Remark 2.9
  • Proposition 2.10
  • ...and 12 more