Binary quadratic refinement ($Q_v(x)=Q_0(x)+B(v,x)$)

A binary quadratic refinement of a fixed nonsingular alternating bilinear form $B$ has polar form $B$. The difference of two refinements is linear, so nondegeneracy gives the displayed unique parametrization by $v$. For $\dim V=2m$, refinements of plus and minus Arf invariant of a quadratic form have respectively $2^{m-1}(2^m+1)$ and $2^{m-1}(2^m-1)$ elements. Completing the square in the binary character sum gives $ϵ(Q_v)=ϵ(Q_0)(-1{)}^{Q_0(v)}$.

The symplectic group over a finite field is transitive on each of the two quadratic types. The stabilizer of a refinement is its orthogonal group over a finite field. In the displayed parametrization, other refinements of the same type correspond to nonzero singular vectors, and refinements of opposite type to nonsingular vectors. Witt's lemma makes the orthogonal stabilizer transitive on each of these vector sets. Thus both type actions are two-transitive for $m\ge 2$, and a minus stabilizer is transitive on the plus orbit. For $m=1$ the minus orbit is a singleton.