Codes and caps from orthogonal Grassmannians

Abstract

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.Comment: Keywords: Polar Grassmannian; dual polar space; embedding; error correcting code; cap; Hadamard matrix; Sylvester construction (this is a slightly revised version of v2, with updated bibliography