Binary codes of the symplectic generalized quadrangle of even order

Let q be a prime power and W ( q ) be the symplectic generalized quadrangle of order q . For q even, let O (respectively, E , T ) be the binary linear code spanned by the ovoids (respectively, elliptic ovoids, Tits ovoids) of W ( q ) and Γ 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 A ∈ { E , T , O } , we describe the codewords of minimum and maximum weights in A and its dual A ⊥ , and show that A is a one-step completely orthogonalizable code (Theorem 1.1 ). We prove that, for q > 2 , any blocking set of P G ( 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 P G ( 3 , q ) (Theorem 1.3 ). We deduce that a clique in Γ has size at most q (Theorem 1.4 ).