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


For a subset ψ of PG(N, 2) a known result states that ψ has polynomial degree ≤ r, rN, if and only if ψ 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 ψ 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 <8 the polynomial degree of \(\mathcal{G}_{1,n,2}\) is \(\left( {\begin{array}{l} n \\ 2 \\ \end{array}} \right) - 1\).

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Communicated by: D. Jungnickel

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Shaw, R., Gordon, N.A. The Polynomial Degree of the Grassmannian \({\mathcal G_{\bf 1,}{\bf n,}{\bf 2}}\). Des Codes Crypt 39, 289–306 (2006).

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  • polynomial degree
  • subsets of PG(N, 2)
  • Grassmannian G1, n, 2

AMS Classification

  • 51E20
  • 05C90
  • 11G25
  • 14M15