The Polynomial Degree of the Grassmannian $${\mathcal G_{\bf 1,}{\bf n,}{\bf 2}}$$

Authors:
R. Shaw;N. A. Gordon
Affiliations:
Department of Mathematics, University of Hull, Hull, United Kingdom HU6 7RX;Department of Computer Science, University of Hull, Hull, United Kingdom HU6 7RX
Venue:
Designs, Codes and Cryptography
Year:
2006

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Abstract

For a subset 驴 of PG(N, 2) a known result states that 驴 has polynomial degree 驴 r, r驴 N, if and only if 驴 intersects every r-flat of PG(N, 2) in an odd number of points. Certain refinements of this result are considered, and are then applied in the case when 驴 is the Grassmannian $$\mathcal{G}_{1,n,2}\subset PG(N, 2), N = \left( {\begin{array}{l} {n + 1} \\ 2 \\ \end{array} } \right) - 1$$ , to show that for n $$\mathcal{G}_{1,n,2}$$ is $$\left( {\begin{array}{l} n \\ 2 \\ \end{array}} \right) - 1$$ .