Codes and caps from orthogonal Grassmannians

TL;DR

This paper explores the properties of linear error-correcting codes and projective caps derived from the orthogonal Grassmannian's Grassmann embedding, providing parameter determinations and identifying special caps.

Contribution

It determines parameters of codes from the orthogonal Grassmannian and identifies certain point sets as projective caps under the Grassmann embedding.

Findings

Parameters of codes from $ ext{Grassmann}$ embedding are established.

Certain point sets form projective caps under the embedding.

The study links geometric structures with coding theory applications.

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.