We show that any PG(2n, q2) is a disjoint union of (q2n+1 − 1)/ (q − 1) caps, each cap consisting of (q2n+1 + 1)/(q + 1) points. Furthermore, these caps constitute the “large points” of a PG(2n, q), with the incidence relation defined in a natural way.
A square matrix H = (hij) over the finite field GF(q2), q a prime power, is said to be Hermitian if hijq = hij for all i, j [1, p. 1161]. In particular, hii ∈ GF(q). If if is Hermitian, so is p(H), where p(x) is any polynomial with coefficients in GF(q).
Given a Desarguesian Projective Geometry PG(2n, q2), n > 0, we denote its points by column vectors:
All Hermitian matrices in this paper will be 2n + 1 by 2n + 1, n > 0.