First I want to review some concept from quadratic form. Let $V$ be quadratic space over finite field $F$ and $char(F)\neq 2$ with quadratic form $q$. For exmaple

$q:V\rightarrow V$ and $|F|=q$ and $\dim V=n$.

We may assume the vectors of $V$ are represented as column vectors $v=(a_1,b_1,…)\in V$, and that $q$ takes one of the three forms

1.$q(v)={a_1}^2-{b_1}^2+…+{a_k}^2-{b_k}^2$

2.$q(v)={a_1}^2-{b_1}^2+…+{a_k}^2-{db_k}^2$

3.$q(v)={a_1}^2-{b_1}^2+…+{a_k}^2-{b_k}^2-{a_{k+1}}^2$

or if we want to have matrix representation we have

1.$q=diag(1,-1,1,-1,…,1,-1)$

2.$q=diag(1,-1,1,-1,…,1,-1,1)$

3.$q=diag(1,-1,1,-1,…,1,-1,1,-d)$

Finally for subspace $W$ we have $W^{\perp}=\{v\in V\mid b(v,w)=0 ,\forall w \in W \}$ where $b(x,y)=(\frac{1}{2})[q(x+y)-q(x)-q(y)]$.

The problem I am interested in :

Compute the number of $k$-dimensional subspaces $W$ such that $W\subseteq W^{\perp}$; this subspace is called **totally isotropic** subspace (T,I). For $k=1$, it is prominent theorem. That comes back to find solution of these equations

1.$q(v)={x_1}^2-{y_1}^2+…+{x_k}^2-{y_k}^2=0$

2.$q(v)={x_1}^2-{y_1}^2+…+{x_k}^2-{dy_k}^2=0$

3.$q(v)={x_1}^2-{y_1}^2+…+{x_k}^2-{y_k}^2-{x_{k+1}}^2=0$ So the number of 1-dimensional (T,I) is

1.$\frac{q^{2k-1}+q^{k}-q^{k-1}-1}{q-1}$

2.$\frac{q^{2k-1}-q^{k}+q^{k-1}-1}{q-1}$

3.$\frac{q^{2k}-1}{q-1}$

Now what can I do for $k\geq 2$.