The maximum size of a partial spread in is

In this paper, we show that the largest maximal partial spreads of the hermitian variety H ( 5 , q 2 ) consist of q 3 + 1 generators. Previously, it was only known that q 4 is an upper bound for the size of these partial spreads. We also show for q ⩾ 7 that every maximal partial spread of H ( 5 , q 2 ) contains at least 2 q + 3 planes. Previously, only the lower bound q + 1 was known.