New bounds for partial spreads of [formula omitted] and partial ovoids of the Ree–Tits octagon

Abstract

Two results are obtained that give upper bounds on partial spreads and partial ovoids respectively.The first result is that the size of a partial spread of the Hermitian polar space H(3,q2) is at most ((2p3+p)/3)t+1, where q=pt, p is a prime. For fixed p this bound is in o(q3), which is asymptotically better than the previous best known bound of (q3+q+2)/2. Similar bounds for partial spreads of H(2d−1,q2), d even, are given.The second result is that the size of a partial ovoid of the Ree–Tits octagon O(2t) is at most 26t+1. This bound, in particular, shows that the Ree–Tits octagon O(2t) does not have an ovoid.