# Residual $q$-Fano Planes and Related Structures

## TL;DR

This paper introduces a new definition for residual $q$-designs, proves the existence of residual $q$-Fano planes for all prime power $q$, and advances understanding of $q$-analog Steiner systems.

## Contribution

It presents a novel definition for residual $q$-designs that better reflects their properties and proves the existence of residual $q$-Fano planes for all prime powers.

## Key findings

- Existence of residual $q$-Fano plane for all prime power $q$
- New definition of residual $q$-designs that aligns with $q$-analog properties
- Progress towards constructing a $q$-Fano plane

## Abstract

One of the most intriguing problems, in $q$-analogs of designs, is the existence question of an infinite family of $q$-analog of Steiner systems, known also as $q$-Steiner systems, (spreads not included) in general, and the existence question for the $q$-analog of the Fano plane, known also as the $q$-Fano plane, in particular. These questions are in the front line of open problems in block design. There was a common belief and a conjecture that such structures do not exist. Only recently, $q$-Steiner systems were found for one set of parameters. In this paper, a definition for the $q$-analog of the residual design is presented. This new definition is different from previous known definition, but its properties reflect better the $q$-analog properties. The existence of a design with the parameters of the residual $q$-Steiner system in general and the residual $q$-Fano plane in particular are examined. We prove the existence of the residual $q$-Fano plane for all $q$, where $q$ is a prime power. The constructed structure is just one step from a construction of a $q$-Fano plane.