Designs, Codes and Cryptography

, Volume 79, Issue 1, pp 163–170 | Cite as

Binary codes of the symplectic generalized quadrangle of even order

Article

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).

Keywords

Generalized quadrangle Ovoid Blocking set Linear code 

Mathematics Subject Classification

51E12 94B05 51E21 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesNational Institute of Science Education and ResearchBhubaneswarIndia
  2. 2.Statistics and Mathematics UnitIndian Statistical InstituteBangaloreIndia

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