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).