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

Abstract

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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References

  1. 1.

    E. F. Assmus J. D. Key (1993) Designs and their Codes Cambridge University Press Cambridge

    Google Scholar 

  2. 2.

    W. Bosma J. Cannon C. Playoust (1997) ArticleTitleThe MAGMA algebra system I: The user language Journal of Symbol Computation 24 235–265 Occurrence Handle1484478

    MathSciNet  Google Scholar 

  3. 3.

    B. N. Cooperstein (1998) ArticleTitleExternal flats to varieties in PG2V) over finite fields Geometriae Dedicata 69 223–235 Occurrence Handle10.1023/A:1005053409486 Occurrence Handle0893.51010 Occurrence Handle99c:51010

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    David G. Glynn, Johannes G. Maks and L. R. A. (Rey) Casse, The polynomial degree of the Grassmannian \(\mathcal{G} (n,1,q)\) of lines in finite projective space PG(n,q), preprint (July 2003).

  5. 5.

    N. A. Gordon T. M. Jarvis J. G. Maks R. Shaw (1994) ArticleTitleComposition algebras and PG(m,2) Journal of Geometry 51 50–59 Occurrence Handle10.1007/BF01226856 Occurrence Handle95h:17002

    Article  MathSciNet  Google Scholar 

  6. 6.

    J. W. P. Hirschfeld (1985) Finite Projective Spaces of Three Dimensions Clarendon Oxford

    Google Scholar 

  7. 7.

    J. W. P. Hirschfeld and R. Shaw, Projective geometry codes over prime fields, see AMS Contemporary Mathematics Series, Vol. 168, G. Mullen and P. J.-S. Shiue (eds.), Finite Fields: Theory, Applications and Algorithms, Amer. Math. Soc. (1994) pp. 151–163.

  8. 8.

    R. Shaw (1992) ArticleTitleA characterization of the primals in PG(m,2) Designs, Codes and Cryptography 2 253–256 Occurrence Handle10.1007/BF00141969 Occurrence Handle0759.51007

    Article  MATH  Google Scholar 

  9. 9.

    R. Shaw, Finite geometries and Clifford algebras III, In A. Micali et al. (eds), Proc. of the 2nd Workshop on Clifford Algebras and their Applications in Mathematical Physics, Montpellier, France, (1989); Kluwer Acad. Pubs. (1992) pp. 121–132.

  10. 10.

    R. Shaw, Composition algebras, PG(m,2) and non-split group extensions, In M. A. del Olmo et al. (eds), Proc. of XIXth International Colloquium on Group Theoretical Methods in Physics, Salamanca (1992), Anales de Fisica, Monografias, Vol. 1, CIEMAT / RSEF, Madrid (1993) pp. 467–470.

  11. 11.

    R. Shaw N. A. Gordon (1994) ArticleTitleThe lines of PG(4,2) are the points of a quintic in PG(9,2) J. Combin. Theory (A) 68 226–231 Occurrence Handle95h:51023

    MathSciNet  Google Scholar 

  12. 12.

    R. Shaw J. G. Maks N. A. Gordon (2005) ArticleTitleThe classification of flats in PG(9,2) which are external to the Grassmannian \(\mathcal{G}_{1,4,2}\) Designs Codes Cryptography 34 203–227 Occurrence Handle2005m:51011

    MathSciNet  Google Scholar 

  13. 13.

    L. H. Soicher, The Design Package for GAP, http://designtheory.org/software/gap_design/.

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Correspondence to R. Shaw.

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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). 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