Dimensional doubly dual hyperovals and bent functions (Q2016997)

Let \(V\) be an \(m\)-dimensional vector space over the finite field \(\mathbb F_q\). A set \(\mathbf{D}\) of \(n\)-dimensional subspaces of \(V\) is called a dual hyperoval of rank \(n\), if \(|\mathbf D| =(q^n-1)/(q-1)+1\) and \(\dim (X\cap Y)=1\), \(X\cap Y\cap Z=0\) holds for any three \(X,Y,Z\in\mathbf D\). If there is a subspace \(W\) such that \(V=X\oplus W\) for all \(X\in\mathbf D\), one says that \(\mathbf D\) splits over \(W\). A dual hyperoval \(\mathbf D\) of rank \(n\) is called doubly dual, if \(m=2n\) and \(\mathbf D\) is also a dual hyperoval of the dual space, i.e., if \(\dim (X+ Y)=2n-1\), \(X+ Y+ Z=V\) holds for any three \(X,Y,Z\in\mathbf D\). Consider now the case that \(m=2n\) and \(\mathbb F_q=\mathbb F_2\). Then \(f:V\to \mathbb F_2\) is called a bent function, if its support is a difference set in \(V\). The main theorem of the paper says that a dual hyperoval of rank \(n\) in \(V\) is doubly dual, if and only if the characteristic function of \((\bigcup\limits_{S\in\mathbf D} S)\setminus \{0\}\) is bent. If \(\mathbf D\) splits over \(W\), then \(\mathbf D\) is doubly dual, if and only if the characteristic function of \(W\cup (\bigcup\limits_{S\in\mathbf D} S)\) is bent. Moreover, in the paper some old and some new examples of doubly dual hyperovals of rank \(n\) and the associated bent functions are studied.