Infinite families of $ 3 $ -designs from o-polynomials
Cunsheng Ding, Chunming Tang · Advances in Mathematics of Communications · 2020
A classical approach to constructing combinatorial designs is the group action of a \begin{document}$ t $\end{document} -transitive or \begin{document}$ t $\end{document} -homogeneous permutation group on a base block, which yields a \begin{document}$ t $\end{document} -design in general. It is open how to use a \begin{document}$ t $\end{document} -transitive or \begin{document}$ t $\end{document} -homogeneous permutation group to construct a \begin{document}$ (t+1) $\end{document} -design in general. It is known that the general affine group \begin{document}$ {\mathrm{GA}}_1( {\mathrm{GF}}(q)) $\end{document} is doubly transitive on \begin{document}$ {\mathrm{GF}}(q) $\end{document} . The classical theorem says that the group action by \begin{document}$ {\mathrm{GA}}_1( {\mathrm{GF}}(q)) $\end{document} yields \begin{document}$ 2 $\end{document} -designs in general. The main objective of this paper is to construct \begin{document}$ 3 $\end{document} -designs with \begin{document}$ {\mathrm{GA}}_1( {\mathrm{GF}}(q)) $\end{document} and o-polynomials. O-polynomials (equivalently, hyperovals) were used to construct only \begin{document}$ 2 $\end{document} -designs in the literature. This paper presents for the first time infinite families of \begin{document}$ 3 $\end{document} -designs from o-polynomials (equivalently, hyperovals).