A Characterization of Hermitian Varieties as codewords

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 >= 4.