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.

Typo in 2.1.1 corrected

References in corpus (17)

• Projective Representations I. Projective lines over rings
• On bijections that preserve complementarity of subspaces
• The Connected Components of the Projective Line over a Ring
• Divisible Designs, Laguerre Geometry, and Beyond
• Extending the Concept of Chain Geometry
• A Classification of the Projective Lines over Small Rings
• Radical parallelism on projective lines and non-linear models of affine spaces
• Jordan homomorphisms and harmonic mappings
• The Dual of a Chain Geometry
• A Classification of the Projective Lines over Small Rings II. Non-Commutative Case
• Some Constructions of Divisible Designs from Laguerre Geometries
• Projective Representations II. Generalized chain geometries
• From pentacyclic coordinates to chain geometries, and back
• A Three-Dimensional Laguerre Geometry and Its Visualization
• Lifting of divisible designs
• Divisible designs from twisted dual numbers
• Affine Circle Geometry over Quaternion Skew Fields

Cited by in corpus (12)

• Projective Ring Line Encompassing Two-Qubits
• Projective Ring Line of an Arbitrary Single Qudit
• Projective Ring Line of a Specific Qudit
• Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators
• Divisible Designs, Laguerre Geometry, and Beyond
• A Classification of the Projective Lines over Small Rings
• From pentacyclic coordinates to chain geometries, and back
• On the Fine Structure of the Projective Line Over GF(2) x GF(2) x GF(2)
• Divisible designs from twisted dual numbers
• Von Staudt's theorem revisited
• Affine and Projective Planes Linked with Projective Lines over Certain Rings of Lower Triangular Matrices
• Linear sets in the projective line over the endomorphism ring of a finite field