Classification of spreads of Tits quadrangles of order 64

Brown et al. provide a representation of a spread of the Tits quadrangle T2(O), O an oval of PG(2,q), q even, in terms of a certain family of q ovals of PG(2,q). By combining this representation with the Vandendriessche classification of hyperovals in PG(2,64) and the classification of flocks of the quadratic cone in PG(3, 64), recently given by the authors, in this paper, we classify all the spreads of T2(O), O an oval of PG(2, 64), up to equivalence. These complete the classification of spreads of T2(O) for q ≤ 64.