Consider a vector space of dimension 2n, \(n \ge 2\) , defined over the finite field of order q, that is equipped with a nondegenerate alternating bilinear form f. Denote by \(W(2n-1,q)\) the symplectic polar space associated with (V, f). For q even, let \({\mathcal {H}}\) denote the binary linear code spanned by those hyperbolic quadrics of \(\textrm{PG}(2n-1,q)\) with quadratic forms \(\kappa \) for which the associated symmetric bilinear form \(f_{\kappa }\) equals f, up to a nonzero factor. The dimension of the code \({\mathcal {H}}\) is known, see Sastry and Sin (J Comb Theory Ser A 94(1):1–14, 2001). In this paper, we characterize the codewords of minimum and maximum weights in \({\mathcal {H}}\) and its dual code \({\mathcal {H}}^{\perp }\) (Theorem 1.1). For all q, we also determine the minimum size blocking sets in \(\textrm{PG}(2n-1,q)\) with respect to the hyperbolic lines of \(W(2n-1,q)\) (Theorem 4.8) which is needed to characterize the maximum weight codewords of \({\mathcal {H}}\) .
