Bilinear dual hyperovals in the largest ambient spaces
Let \(U\) be a vector space over a finite field \(\mathbb F_q\) (\(Q\) a prime power). A dual hyperoval (DHO) of rank \(n\) over \(\mathbb F_q\) is a set \(\mathcal S\) of \(n\)-dimensional subspaces of \(U\) fulfilling the following properties:{\parindent=0.7cm \begin{itemize}\item[(i)] \(|\mathcal S|=\frac{q^n-1}{q-1}+1\), \item[(ii)] \(\dim(X\cap Y)=1\) and \(X\cap Y\cap Z \cap Z = \{0\}\) for any pairwise distinct \(X\), \(Y\), \(Z\) of \(\mathcal S\). \end{itemize}} The subspace \(A(\mathcal S)\) of \(U\) spanned by all the members of \(\mathcal S\) is the ambient space of \(\mathcal S\). A DHO \(\mathcal S\) of rank \(n\) over \(\mathbb F_2\) is bilinear if there are spaces \(X\) and \(Y\) over \(\mathbb F_2\) with \(\dim X = n\) and an \(\mathbb F_2\)-bilinear map \(\beta\) from \(X\times Y\) into \(Y\) such that \(\mathcal S=\{X(e)\mid e\in X\}\), where \(X(e)\) is a subspace of \(X\otimes Y=\{(x,y)\mid x\in X,y\in Y\}\) defined by \[ x(e)= \{(x, (x, e)\beta)\mid x\in X\}. \] In this paper, the following facts are established: the conjecture \(\dim A(\mathcal S)\leq\frac{n(n+1)}{2}\) is true for all bilinear DHO of rank \(n\), and \(\dim A(\mathcal S) =\frac{n(n+1)}{2}\) for a bilinear DHO \(\mathcal S\) of rank \(n\) if and only if \(\mathcal S\) is isomorphic to the Huybrechtes or the Buratti-Del Fra DHO.
scientific article; zbMATH DE number 6820925
scientific article