Isomorphisms and automorphisms of extensions of bilinear dimensional dual hyperovals and quadratic APN functions (Q254322)

The authors consider their previously defined construction that transforms any symmetric, bilinear \(n\)-dimensional dual hyperoval over the field with two elements into an \((n+1)\)-dimensional dual hyperoval. This construction is naturally related to a construction of almost perfect nonlinear functions in dimension \(n+1\) using two such functions in dimension \(n\). The authors show that the isomorphism problem, and similarly the automorphism problem, for the \((n+1)\)-dimensional extension can be reduced to the isomorphism problem of the initial \(n\)-dimensional objects. They introduce the extension group of a dimensional dual hyperoval which characterizes those dimensional dual hyperovals that are extensions. Additionally, they show that the extension construction is the source of large numbers of inequivalent almost perfect nonlinear functions of degree three.