  In this paper we investigate linear error correcting codes and projective
caps related to the Grassmann embedding $\varepsilon_k^{gr}$ of an orthogonal
Grassmannian $\Delta_k$. In particular, we determine some of the parameters of
the codes arising from the projective system determined by
$\varepsilon_k^{gr}(\Delta_k)$. We also study special sets of points of
$\Delta_k$ which are met by any line of $\Delta_k$ in at most 2 points and we
show that their image under the Grassmann embedding $\varepsilon_k^{gr}$ is a
projective cap.
