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

Designs, Codes and Cryptography volume 39, pages289–306(2006)Cite this article

Abstract

For a subset ψ of PG(N, 2) a known result states that ψ has polynomial degree ≤ r, r≤ N, 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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Affiliations

  1. Department of Mathematics, University of Hull, Hull, HU6 7RX, United Kingdom

    R. Shaw

  2. Department of Computer Science, University of Hull, Hull, HU6 7RX, United Kingdom

    N. A. Gordon

Authors
  1. R. Shaw
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  2. N. A. Gordon
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Correspondence to R. Shaw.

Additional information

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). https://doi.org/10.1007/s10623-005-4524-4

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Keywords

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

AMS Classification

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