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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]

By Akihiro Munemasa.

Verified locator: Author lecture notes based on 1994 and 1995 lectures (preface). Section 2, initial definition and Proposition 2.1; Theorem 2.11 classifies nondegenerate even-dimensional forms by Witt indices m or m−1.