  Let Q_0 be the classical generalized quadrangle of order q = 2n arising from
a non-degenerate quadratic form in a 5-dimensional vector space defined over a
finite field of order q. We consider the rank two geometry X having as points
all the elliptic ovoids of Q_0 and as lines the maximal pencils of elliptic
ovoids of Q_0 pairwise tangent at the same point. We first prove that X is
isomorphic to a 2-fold quotient of the affine generalized quadrangle Q
\setminus Q_0 where Q is the classical (q; q^2)-generalized quadrangle
admitting Q_0 as a hyperplane. Then, we investigate the collinearity graph
\Gamma of X: In particular, we obtain a classification of the cliques of \Gamma
proving that they arise either from lines of Q or subgeometries of Q defined
over F_2

Combinatorics