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