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

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 $1$