# 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).

### Keywords

Generalized quadrangle Ovoid Blocking set Linear code### Mathematics Subject Classification

51E12 94B05 51E21### References

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