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.