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Binary codes of the symplectic generalized quadrangle of even order

Abstract

Let \(q\) be a prime power and \(W(q)\) be the symplectic generalized quadrangle of order \(q\). For \(q\) even, let \(\mathcal {O}\) (respectively, \(\mathcal {E}\), \(\mathcal {T}\)) be the binary linear code spanned by the ovoids (respectively, elliptic ovoids, Tits ovoids) of \(W(q)\) and \(\Gamma \) be the graph defined on the set of ovoids of \(W(q)\) in which two ovoids are adjacent if they intersect at one point. For \(\mathcal {A}\in \{\mathcal {E},\mathcal {T},\mathcal {O}\}\), we describe the codewords of minimum and maximum weights in \(\mathcal {A}\) and its dual \(\mathcal {A}^{\perp }\), and show that \(\mathcal {A}\) is a one-step completely orthogonalizable code (Theorem 1.1). We prove that, for \(q>2\), any blocking set of \(PG(3,q)\) with respect to the hyperbolic lines of \(W(q)\) contains at least \(q^2+q+1\) points and equality holds if and only if it is a hyperplane of \(PG(3,q)\) (Theorem 1.3). We deduce that a clique in \(\Gamma \) has size at most \(q\) (Theorem 1.4).

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Acknowledgments

The first author would like to thank the Indian Statistical Institute, Bangalore Center, for the kind hospitality provided during his visit to the Statistics and Mathematics Unit in the summer of 2014.

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Correspondence to Binod Kumar Sahoo.

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Communicated by J. H. Koolen.

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Sahoo, B.K., Sastry, N.S.N. Binary codes of the symplectic generalized quadrangle of even order. Des. Codes Cryptogr. 79, 163–170 (2016). https://doi.org/10.1007/s10623-015-0040-3

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  • DOI: https://doi.org/10.1007/s10623-015-0040-3

Keywords

  • Generalized quadrangle
  • Ovoid
  • Blocking set
  • Linear code

Mathematics Subject Classification

  • 51E12
  • 94B05
  • 51E21