Nonlinear hyperplane modifications of the Gold APN function

Determine whether APN functions of the form G(x)=x^3+Tr(x)Q(x) over F_{2^n} exist when Q is quadratic or, more generally, nonlinear, extending the paper’s characterization of linearized polynomials L satisfying the APN condition in equation (3.1).

Background

Theorem 3.7 characterizes the APN property of functions x^3+Tr(x)L(x) using an exponential-sum condition when L is linearized. The conclusion asks for further linearized polynomials satisfying that condition and then raises the unresolved possibility of replacing the linear mapping L by a quadratic or more general nonlinear function Q.

A positive answer would extend the hyperplane-modification construction beyond quadratic APN functions generated by linear perturbations and could yield broader families of APN mappings.