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Designs, Codes and Cryptography

, Volume 87, Issue 4, pp 909–929 | Cite as

Three homogeneous embeddings of \(\textit{DW}(2n-1,2)\)

  • Bart De BruynEmail author
Article
  • 13 Downloads
Part of the following topical collections:
  1. Special Issue: Finite Geometries

Abstract

We construct a homogeneous full projective embedding of the dual polar space \(\textit{DW}(2n-1,2)\) from the hyperplane intersections of hyperbolic type of the parabolic quadric Q(2n, 2). We believe that this embedding is universal, but have not succeeded in proving that. As a by-product of our investigations, we have obtained necessary and sufficient conditions for this to be the case and came across two other homogeneous full projective embeddings of \(\textit{DW}(2n-1,2)\), one with vector dimension \(\frac{2^{2n-1} + 3 \cdot 2^{n-1} -2}{3}\) and another one with vector dimension \(\frac{2^{2n-1} + 3 \cdot 2^{n-1} -2 - 6n}{3}\).

Keywords

Symplectic dual polar space Homogeneous projective embedding Universal projective embedding 

Mathematics Subject Classification

51A45 51A50 05B25 20C33 

Notes

Acknowledgements

The author wishes to thank Peter Vandendriessche for performing the computer computations which showed that the hyperbolic embedding \(\epsilon _h\) of \(\textit{DW}(2n-1,2)\) is universal for \(n \le 7\). He also wishes to thank Bert Seghers for discussions on the topic of this paper.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsGhent UniversityGhentBelgium

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