Let \(\mathbb{Z}_ k\) denote the integers modulo \(k\). The mapping \(j: \mathbb{Z}_ 2 \to \mathbb{Z}_ 4: 1\to 2\) is a group homomorphism. Let \(V\) be a vector space over \(\mathbb{Z}_ 2\) and let \(B\) be a bilinear form on \(V\). A function \(Q:V \to\mathbb{Z}_ 4\) is called a \(\mathbb{Z}_ 4\)-valued quadratic form if \(Q(u+v) = Q(u) + Q(v) + jB(u, v)\) for all \(u, v\) in \(V\). The bilinear form \(B\) is not necessarily alternating.
The author shows that a nonsingular \(\mathbb{Z}_ 4\)-valued quadratic form is determined by its Brown invariant, a generalization of the Arf invariant. He also gives necessary and sufficient conditions for the extension of a \(Q\)-isometry of subspaces to a \(Q\)-isometry of the whole space. Some applications to coding theory follow.