Symmetric doubly dual hyperovals have an odd rank
A set \(\mathcal S\) of \(n\)-dimensional subspaces of a finite-dimensional vector space \(U\) over \(\mathbb F_q\) is a dual hyperoval of rank \(n\) in \(U\), if \(|\mathcal S|=(q^n-1)/(q-1)+1\), any two elements of \(\mathcal S\) meet in a \(1\)-dimensional space, any three elements of \(\mathcal S\) meet in the zero space, and \(\mathcal S\) spans \(U\). In this case, \(\mathcal S\) is called doubly dual if it is also a dual hyperoval of rank \(n\) in the vector space dual to \(U\); then \(\dim U=2n\). Let \(\mathbb F_q=\mathbb F_2\) and let \(U=X\oplus Y\) with \(\dim X=n\). Then, the dual hyperoval \(\mathcal S\) is called bilinear, if there is a monomorphism \(\beta: X\to \mathrm{Hom}(X,Y)\) such that \(\mathcal S=\mathcal S_\beta:=\{S_e\mid e\in X\}\), where \(S_e:=\{(x,x\beta(e)\mid x\in X\}\). The bilinear dual hyperoval \(\mathcal S_\beta\) is symmetric if \(\beta=\beta^o:X\to \mathrm{Hom}(X,Y)\) with \(x\beta^o(e):=e\beta(x)\), and alternating if \(e\beta(e)=0\) for all \(e\in X\). Consider the special case \(Y=X\). Then, \(\mathcal S_\beta\) may be doubly dual. Fix a non-degenerate bilinear form \((\, , \,)\) on \(X\) and let \(\beta^t:X\to \mathrm{Hom}(X,X)=\mathrm{End}(X): e\mapsto \beta(e)^t\), where \(\beta(e)^t\) is the adjoint of \(\beta(e)\) with respect to \((\, , \,)\). The mappings \(\beta\mapsto\beta^o\) and \(\beta\mapsto\beta^t\) are called the Knuth operations. \textit{Y. Edel} [``On some representations of quadratic APN functions and dimensional dual hyperovals'', RIMS Kôkyûroku 1687, 118--130 (2010)] used the Knuth operations in order to prove one direction of \textit{H. Taniguchi}'s theorem [Finite Fields Appl. 15, No. 6, 673--681 (2009; Zbl 1185.51012)], which says that an alternating dual hyperoval is doubly dual if, and only if, it has odd rank. In the present paper, also the other direction of this theorem is proved using the Knuth operations. Moreover, also the following theorem is proved by applying this tool: A symmetric doubly dual hyperoval has odd rank.
scientific article; zbMATH DE number 6390336