On the Generation of Some Dual Polar Spaces of Symplectic Type OverGF(2)

It is demonstrated that the dual polar space of typeSp(2n,2) can be generated as a geometry by λ(n) =(2n+ 1)(2n− 1+ )3points whenn= 4 and 5. In the latter case this affirmatively resolves a conjecture of Brouwer that the dimension of the universal projective embedding of this geometry is λ(5).