The first example of a simple $2-(81,6,2)$ design

TL;DR

This paper presents the first known example of a simple 2-(81,6,2) combinatorial design, constructed using elementary abelian groups and geometric structures over finite fields, demonstrating a novel intersection of algebra and geometry.

Contribution

It provides the first explicit example of a simple 2-(81,6,2) design, combining group theory and geometric methods in finite fields.

Findings

First explicit example of a simple 2-(81,6,2) design

Uses elementary abelian group of order 81

Constructed from geometric structures over finite fields

Abstract

We give the very first example of a simple $2-(81,6,2)$ design. Its points are the elements of the elementary abelian group of order 81 and each block is the union of two parallel lines of the 4-dimensional geometry over the field of order 3. Hence it is also additive.