Distance regular graphs arising from dimensional dual hyperovals
Let \(m\) and \(n\) be integers with \(m > n \geq 3\). Let \(\mathbb{F}_q\) be the finite field of \(q\) elements and \(U = V (m, q)\) be an \(m\)-dimensional \(\mathbb{F}_q\)-vector space. A family \(\mathcal{S}\) of \(n\)-dimensional vector subspaces in \(U\) is called an \(n\)-dimensional dual hyperoval if it satisfies the following conditions \begin{itemize} \item[(D1)] any two distinct members of \(\mathcal{S}\) intersect at a 1-dimensional subspace; \item[(D2)] any three mutually distinct members of \(\mathcal{S}\) intersect trivially; \item[(D3)] the union of the members of \(\mathcal{S}\) generates \(U\); \item[(D4)] \(\#\mathcal{S}=q^{n-1}+q^{n-2}+\cdots q+2\). \end{itemize} Consider \(q=2^n\). A dual hyperoval \(\mathcal{S}\) can be described as \(\mathcal{S}_B:=\{(x,B(x,t)) \mid x,t \in \mathbb{F}_q\}\) if and only if \(B(x,a)=0\) and \(B(b,x)=0\) have exactly one solution for each \(a,b \in \mathbb{F}_q^*\) and \(B\) is a \(\mathbb{F}_2\)-bilinear form. It is possible to define another incidence structure \(\mathcal{S}_B^{\bot}\) using the inner product \((x,a)\cdot (y,b)=Tr(xb+ya)\), where \(Tr\) is the absolute trace from \(\mathbb{F}_q\) to \(\mathbb{F}_2\). A doubly dual hyperoval \(\mathcal{S}_B\) is a dual hyperoval such that \(\mathcal{S}_B^{\bot}\) is also a dual hyperoval. To each dual hyperoval \(\mathcal{S}\) a particular incidence structure called semibiplane \(\Gamma(\mathcal{S}_B)\) can be associated. In this paper, a result of \textit{A. Pasini} and \textit{S. Yoshiara} [Eur. J. Comb. 22, No. 4, 547--560 (2001; Zbl 1004.51017)] concerning the incidence of specific graphs \(\Gamma(\mathcal{S}_B)\) is extended. In particular, the main result is the following. Theorem. The incidence graph of \(\Gamma(\mathcal{S}_B)\) is a distance regular graph if \(\mathcal{S}_B\) is a doubly dual hyperoval.
scientific article; zbMATH DE number 7313139
scientific article