Binary codes of the symplectic generalized quadrangle of even order

Abstract

Let $$q$$q be a prime power and $$W(q)$$W(q) be the symplectic generalized quadrangle of order $$q$$q. For $$q$$q even, let $$\mathcal {O}$$O (respectively, $$\mathcal {E}$$E, $$\mathcal {T}$$T) be the binary linear code spanned by the ovoids (respectively, elliptic ovoids, Tits ovoids) of $$W(q)$$W(q) and $$\Gamma $$Γ be the graph defined on the set of ovoids of $$W(q)$$W(q) in which two ovoids are adjacent if they intersect at one point. For $$\mathcal {A}\in \{\mathcal {E},\mathcal {T},\mathcal {O}\}$$A?{E,T,O}, we describe the codewords of minimum and maximum weights in $$\mathcal {A}$$A and its dual $$\mathcal {A}^{\perp }$$A?, and show that $$\mathcal {A}$$A is a one-step completely orthogonalizable code (Theorem 1.1). We prove that, for $$q>2$$q>2, any blocking set of $$PG(3,q)$$PG(3,q) with respect to the hyperbolic lines of $$W(q)$$W(q) contains at least $$q^2+q+1$$q2+q+1 points and equality holds if and only if it is a hyperplane of $$PG(3,q)$$PG(3,q) (Theorem 1.3). We deduce that a clique in $$\Gamma $$Γ has size at most $$q$$q (Theorem 1.4).