Design Lines

TL;DR

This paper explores the geometric structure of block designs by analyzing the lines within a three-dimensional affine variety that characterizes their parameters, revealing organizational properties and examples of design families.

Contribution

It introduces a geometric perspective on block designs through the study of lines in an affine variety, providing new insights into their structure and classification.

Findings

Lines in the affine variety organize classes of designs.

Examples of design families following these lines are presented.

The geometric approach offers a new perspective on design properties.

Abstract

The two basic equations satisfied by the parameters of a block design define a three-dimensional affine variety $\mathcal{D}$ in $\mathbb{R}^{5}$. A point of $\mathcal{D}$ that is not in some sense trivial lies on four lines lying in $\mathcal{D}$. These lines provide a degree of organization for certain general classes of designs, and the paper is devoted to exploring properties of the lines. Several examples of families of designs that seem naturally to follow the lines are presented.