# Subspaces intersecting in at most a point

## TL;DR

This paper improves the lower bounds on the maximum number of planes in a projective space that intersect in at most a point, using a new construction strategy with implications for constant dimension codes.

## Contribution

It introduces a new construction method that enhances bounds on plane arrangements in projective spaces and constant dimension codes.

## Key findings

- New lower bound for plane arrangements in PG(8,q)
- Enhanced bounds for constant dimension codes A_q(9,4;3)
- General construction strategy applicable to related problems

## Abstract

We improve on the lower bound of the maximum number of planes in $\operatorname{PG}(8,q)\cong\F_q^{9}$ pairwise intersecting in at most a point. In terms of constant dimension codes this leads to $A_q(9,4;3)\ge q^{12}+ 2q^8+2q^7+q^6+2q^5+2q^4-2q^2-2q+1$. This result is obtained via a more general construction strategy, which also yields other improvements.