Two-Character Sets as Subsets of Parabolic Quadrics

Abstract

A two-character set is a set of points of a finite projective space that has two intersection numbers with respect to hyperplanes. Two-character sets are related to strongly regular graphs and two-weight codes. In the literature, there are plenty of constructions for (non-trivial) two-character sets by considering suitable subsets of quadrics and Hermitian varieties. Such constructions exist for the quadrics \(Q^{+}(2n-1,4) \subseteq PG(2n-1,q)\), \(Q^{-}(2n+1,4) \subseteq PG(2n+1,q)\) and the Hermitian varieties \(H(2n-1,q^{2}) \subseteq PG(2n-1,q^{2})\), \(H(2n,q^{2}) \subseteq PG(2n,q^{2})\). In this note, we show that every two-character set of \(PG(2n,q)\) that is contained in a given nonsingular parabolic quadric \(Q(2n,q) \subseteq PG(2n,q)\) is a subspace of \(PG(2n,q)\). This offers some explanation for the absence of the parabolic quadrics in the above-mentioned constructions.