On \(d\)-dimensional dual hyperovals
Let \(\Pi=\text{PG}(n,q)\) be the \(n\)-dimensional projective space over the finite field with \(q\) elements. A \(d\)-dimensional dual hyperoval is a family \(F\) of \(d\)-dimensional subspaces of \(\Pi\) such that each point of \(\Pi\) is contained in 0 or 2 elements of \(F\), any two elements of \(F\) intersect in exactly one point, and the set of points belonging to the elements of \(F\) spans \(\Pi\). A \(d\)-dimensional dual hyperoval is said to satisfy property (T), if for all \(U,V,W\in F\), we have \(\dim(\langle U,V\rangle\cap W)=1\). It is said to be \textit{2-transitive}, if its stabilizer in \(\Aut(\Pi)\) acts 2-transitively on \(F\). After giving some examples, parameters and restrictions for \(d\)-dimensional dual hyperovals, the author gives a complete classification of 2-dimensional dual hyperovals in \(\text{PG}(4,2)\) and in \(\text{PG}(5,2)\). This yields a complete classification of 2-dimensional dual hyperovals satisfying property (T) as well as a classification of 2-transitive 2-dimensional dual hyperovals.
scientific article; zbMATH DE number 1436010