# Ovoidal fibrations in $PG(3,q), q$ even

## TL;DR

This paper proves a geometric property of ovoid partitions in projective 3-space over even fields, relating lines and their polars to tangent ovoids, using coding theory and symplectic forms.

## Contribution

It establishes a new relation between lines, their polars, and tangent ovoids in $PG(3,q)$ for even q, utilizing linear code radicals and symplectic geometry.

## Key findings

- Lines and their polars are tangent to distinct ovoids.
- The radical of the associated linear code has codimension 1.
- The geometric configuration is characterized for partitions by ovoids.

## Abstract

We prove that, given a partition of the point-set of $PG(3,q), q=2^n >2$, by ovoids $\{\theta_i\}^q_{i=0}$ of $PG(3,q)$ and a line $\ell$ of $PG(3,q)$, not tangent to $\theta_0$ if $\ell^\perp$ denotes the polar of $\ell$ relative to the symplectic form on $PG(3,q)$ whose isotropic lines are the tangent lines to $\theta_0$, then $\ell$ and $\ell^\perp$ are tangent to distinct ovoids $\theta_j, \theta_k$, both distinct from $\theta_0$. This uses the fact that the radical of the linear code generated by the dual duals $\ell\cup \ell^\perp$ of the hyperbolic quadrics , with $\ell$ and $\ell^\perp$ as above, is of codimension $1$