# $(22_4)$ and $(26_4)$ configurations of lines

## TL;DR

This paper introduces a technique for creating line arrangements with special properties and successfully constructs specific configurations, narrowing down the open cases in geometric line configurations.

## Contribution

The paper presents a new method for constructing line arrangements and explicitly constructs the $(22_4)$ and $(26_4)$ configurations, advancing the understanding of geometric $(n_4)$ configurations.

## Key findings

- Constructed $(22_4)$ configuration
- Constructed $(26_4)$ configuration
- Only the case $n=23$ remains open for geometric $(n_4)$ configurations

## Abstract

We present a technique to produce arrangements of lines with nice properties. As an application, we construct $(22_4)$ and $(26_4)$ configurations of lines. Thus concerning the existence of geometric $(n_4)$ configurations, only the case $n=23$ remains open.