Some doubly transitive bilinear dual hyperovals and their ambient spaces
Throughout let \(n\geq 5\) be an integer, \(r:=2^n-1\), and \(k\in [0, r-1]\cap{\mathbb Z}\) with \(\gcd(k, r)=1\). A dual hyperoval (DHO for short) of rank \(n\) is a collection \(\mathbf D\) of \(r+1\) subspaces of dimension \(n\) of a given finite dimensional vector space \(V\) over the binary field \(\mathbb F\) provided that any two distinct members of \(\mathbf D\) intersects at a \(1\)-dimensional space, and any three different members of \(\mathbf D\) intersect at the zero space. The ambient space \(V_{\mathbf D}\) of \(\mathbf D\) is the linear subspace in \(V\) generated by the members of \(\mathbf D\). A morphism of DHOs structures is a linear map between their ambient spaces. We say that \(\mathbf D\) splits over \(Y\) if \(V=X\oplus Y\) for all \(X\in\mathbf D\). An DHO \(\mathbf D\) is bilinear if there exists an elementary abelian \(2\)-subgroup \(T\) of \(\text{Aut}(\mathbf D)\) such that it acts regularly on \(\mathbf D\) and such that \(\mathbf D\) splits over the centralizer of \(T\) in \(V_{\mathbf D}\). Thus, the normalizer of \(T\) in \(\text{Aut}(\mathbf D)\) is of type \(T\cdot A\) with \(A\) being the stabilizer of some element \(X\) of \(\mathbf D\) [\textit{U. Dempwolff} and \textit{Y. Edel}, J. Algebr. Comb. 39, No. 2, 457--496 (2014; Zbl 1292.05068)]. The isomorphic class of \(A\) does not depend on \(X\) or \(T\) and it is the so-called autotopism group of \(\mathbf D\).\newline The main goal in the paper under review is to generalize results of \textit{S. Yoshiara} [Eur. J. Comb. 29, No. 7, 1521--1534 (2008; Zbl 1147.51007)], namely to show the existence of bilinear dual hyperovals \(\mathbf D\) of rank \(n\) for a given cyclic autotopism group \(A\) of order \(2^n-1\).
scientific article; zbMATH DE number 6370973