On the generation of some dual polar spaces of symplectic type over \(GF(2)\) (Q1372610)

It is shown that the dual polar symplectic spaces of rank 4 and 5 over GF(2) are generated by 51 and 187 points, respectively (generated here means that the smallest subspace containing the points in question is the whole space itself; a subspace is a set of points which meets every line in 0,1 or all points of the line). For rank 2 and 3 this is respectively 5 and 15. Hence, if the rank is \(n\), then the dual of the symplectic space \(\text{Sp} (2n,2)\) is generated by \({(2^n+1) (2^{n-1}+1) \over 3}\) points.