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Akihiro Munemasa connects finite geometry, quadratic forms and algebraic combinatorics. His author-written notes The geometry of orthogonal groups over finite fields by Akihiro Munemasa develop geometric objects and associated graphs from quadratic forms over finite fields. The preface makes the purpose explicit: the geometry supplies nontrivial examples for studying graph symmetries.^[ref]

For a concrete comparison over ${\mathbb{F}}_3$, consider $Q_1(x,y)=xy$ and $Q_2(x,y)=x^2+y^2$. The first vanishes on both coordinate axes, giving nonzero isotropic vectors. The second vanishes only at $(0,0)$: a nonzero square is $1$, and $1+1=2\ne 0$ in ${\mathbb{F}}_3$. Thus equal-dimensional nondegenerate quadratic forms can have different Witt indices, here one and zero. The existence or absence of totally isotropic subspaces changes the incidence geometry built from the form. Munemasa's notes turn these algebraic distinctions into explicit geometric and combinatorial constructions.^[ref]

The Witt index of a nondegenerate quadratic form $q$ on a finite-dimensional vector space $V$ is the largest dimension of a totally isotropic subspace: a linear subspace $U$ for which $q(u)=0$ for every $u\in U$. In characteristic different from two, the Witt decomposition writeswhere $H$ is a hyperbolic plane of a quadratic form, $q_{\mathrm{an}}$ is an anisotropic quadratic form, and $r$ is the Witt index. The anisotropic part has no nonzero vector on which the form vanishes.^[ref]

For the real form $x_1^2+x_2^2-x_3^2-x_4^2$, the subspace spanned by $(1,0,1,0)$ and $(0,1,0,1)$ is totally isotropic. Projection onto the two negative coordinates is injective on any totally isotropic subspace, giving the matching upper bound two. More generally, a real form of signature $(a,b)$ has Witt index $\min (a,b)$.

The base field matters: $x^2+y^2$ has index zero over $\mathbb{R}$ but index one over $\mathbb{C}$, using the line spanned by $(1,i)$. In characteristic two, vanishing of the associated polar form on $U$ alone is insufficient; the condition remains $q{|}_U=0$. Over a finite field, the two nondegenerate types in dimension $2m$ have Witt indices $m$ and $m-1$.^[ref]