On the smallest minimal blocking sets of \(Q(2n,q)\), for \(q\) an odd prime
The authors characterize the smallest blocking sets of \(Q(2n+2,q)\), \(n\geq2\), \(q>3\) a prime. The main theorem of the paper reads as follows: Let \(\mathcal K\) be a minimal blocking set different from an ovoid of \(Q(2n+2,q)\), \(q\) an odd prime, \(q>3\), \(n\geq2\) or \(q=3\), \(n=2\), and \(| {\mathcal K}| \leq q^{n+1}+q^{n-1}\). Then there is an \((n-2)\)-dimensional space \(\pi_{n-2}\subset Q(2n+2,q)\), \(\pi_{n-2}\cap{\mathcal K}=\emptyset\), with the following property: the tangent space on \(\pi_{n-2}\) intersects \(Q(2n+2,q)\) in a cone \(\pi_{n-2}Q(4,q)\) and \(\mathcal K\) is a cylinder \(\pi_{n-2}{\mathcal O}\setminus\pi_{n-2}\), where \(\mathcal O\) is a three-dimensional elliptic quadric of \(Q(4,q)\), and \(| {\mathcal K}| =q^{n+1}+q^{n-1}\). Since it is now known from the results of \textit{S. Ball, P. Govaerts} and \textit{L. Storme} [On ovoids of parabolic quadrics, Des. Codes. Cryptogr., to appear], and \textit{C. M. O'Keefe} and \textit{J. A. Thas} [Eur. J. Comb. 16, No. 1, 87--92 (1995; Zbl 0819.51005)], that for \(q>3\) prime the quadrics \(Q(2n+2,q)\) do not have ovoids, the characterization follows. In the case \(q=3\), \(n>2\), the description is a bit different.
scientific article; zbMATH DE number 2182132
scientific article