Intersection numbers for subspace designs

TL;DR

This paper introduces intersection numbers for subspace designs, provides $q$-analog equations, and explores the implications for the existence of a $q$-analog of the Fano plane, linking it to other subspace designs.

Contribution

It develops $q$-analog equations for intersection numbers and analyzes the structure of potential $q$-analogs of the Fano plane, offering new insights into subspace design theory.

Findings

Determined the intersection structure of a hypothetical $q$-analog of the Fano plane.

Showed that existence of the $q$-analog implies a specific subspace design.

Presented simplified proofs for intersection numbers in classical block designs.

Abstract

Intersection numbers for subspace designs are introduced and $q$-analogs of the Mendelsohn and K\"ohler equations are given. As an application, we are able to determine the intersection structure of a putative $q$-analog of the Fano plane for any prime power $q$. It is shown that its existence implies the existence of a $2$-$(7,3,q^4)_q$ subspace design. Furthermore, several simplified or alternative proofs concerning intersection numbers of ordinary block designs are discussed.