A Characterization of Hermitian Varieties as Codewords

Abstract

It is known that the Hermitian varieties are codewords in the code defined by the points and hyperplanes of the projective spaces $\mathrm{PG}(r,q^2)$. In finite geometry, also quasi-Hermitian varieties are defined. These are sets of points of $\mathrm{PG}(r,q^2)$ of the same size as a non-singular Hermitian variety of $\mathrm{PG}(r,q^2)$, having the same intersection sizes with the hyperplanes of $\mathrm{PG}(r,q^2)$. In the planar case, this reduces to the definition of a unital. A famous result of Blokhuis, Brouwer, and Wilbrink states that every unital in the code of the points and lines of $\mathrm{PG}(2,q^2)$ is a Hermitian curve. We prove a similar result for the quasi-Hermitian varieties in $\mathrm{PG}(3,q^2)$, $q=p^{h}$, as well as in $\mathrm{PG}(r,q^2)$, $q=p$ prime, or $q=p^2$, $p$ prime, and $r\geq 4$.