# Binary codes of the symplectic generalized quadrangle of even order

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Binod Kumar Sahoo

National Institute of Science Education and Research, Bhubaneswar, India

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N. S. Narasimha Sastry

Indian Statistical Institute, Bangalore, India

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**Minisymposium:**
FINITE GEOMETRY

**Content:**
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. 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 $PG(3,q)$. We deduce that a clique in $\Gamma$ has size at most $q$.