Two-character sets as subsets of parabolic quadrics

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, q) subset of PG(2n - 1, q), Q(-)(2n + 1, q) subset of PG(2n + 1, q) and the Hermitian varieties H(2n - 1, q(2)) subset of PG(2n - 1, q(2)), H(2n, q(2)) subset of PG(2n, q2). 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) subset of 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.

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