The sets closest to ovoids in \(Q^-(2n+1,q)\)
An ovoid of a polar space is a set of points with the property that every maximal subspace of the polar space contains exactly one point of the ovoid. It is known by a result of \textit{J. A. Thas} [Geom. Dedicata 10, 135-144 (1981; Zbl 0458.51010)] that the elliptic quadric \(Q^-(2n+1,q)\) does not have an ovoid when \(n \geq 2\). The author reproves and improves this result of Thas by characterizing the smallest subsets of an elliptic quadric \(Q^-(2n+1,q)\) which intersect every maximal subspace of \(Q^-(2n+1,q)\) in at least one point. He proves that these sets \(B\) have cardinality \(q^{n+1}+q^{n-1}\) with equality if and only if \(B=(U^\perp \setminus U) \cap Q^-(2n+1,q)\) for a subspace \(U\) of dimension \(n-2\) with \(U \subset Q^-(2n+1,q)\). This problem can be generalized to the problem of finding the smallest subsets \(B\) of an elliptic quadric \(Q^-(2n+1,q)\) which intersect all the \(s\)-dimensional subspaces, \(s \leq n-1\), contained in \(Q^-(2n+1,q)\) in at least one point. These sets have been characterized completely by the author in an other article: `A Bose-Burton theorem for elliptic polar spaces' which will appear in Des. Codes Cryptography. The work performed in this article and in the second article are part of a global project of the author to characterize the smallest subsets of polar spaces intersecting all the \(s\)-dimensional subspaces contained in these polar spaces.
scientific article; zbMATH DE number 1266821