Small point sets that meet all generators of \(W(2n+1,q)\) (Q1878296)

A set \(O\) of points meeting exactly once every generator of a polar space \(P\) is called an ovoid of \(P\). Clearly, \(O\), if it exists, is a blocking set of minimum cardinality for the set of all generators of \(O\). It has been shown by Thas that several polar spaces, namely \(Q^{-}(2n+1,q)\) with \(n\geq 2\), \(W(2n+1,q)\) with \(q\) even and \(n\geq 2\) or with \(q\) odd and \(n\geq 1\) and \(H(2n,q)\) with \(n\geq 2\), do not admit ovoids. It is then meaningful to ask what is the minimum size of a blocking set for the generators of these spaces. In this paper, the author proves that the size of such a blocking set \(B\) for \(W(2n+1,q)\) with \(q\) even is \( q^{n-1}(q^2+1)\) and characterises exactly the sets meeting this bound as those consisting of the points of a cone with a \((n-2)\)-dimensional vertex over an ovoid of \(W(3,q)\).