A normal form and schemes of quadratic forms

We present a solution of the problem of the construction of a normal diagonal form for quadratic forms over a local principal ideal ring $R=2R$ with a QF-scheme of order 2. We give a combinatorial representation for the number of classes of projective congruence quadrics of the projective space over $R$ with nilpotent maximal ideal. For the projective planes, the enumeration of quadrics up to projective equivalence is given; we also consider the projective planes over rings with nonprincipal maximal ideal.
We consider the normal form of quadratic forms over the field of $p$-adic numbers. The corresponding QF-schemes have order 4 or 8. Some open problems for QF-schemes are mentioned. The distinguished finite QF-schemes of local and elementary types (of arbitrarily large order) are realized as the QF-schemes of a field.