Infinite families of $ 3 $-designs from o-polynomials

Abstract

<p style='text-indent:20px;'>A classical approach to constructing combinatorial designs is the group action of a <inline-formula><tex-math id="M2">\begin{document}$ t $\end{document}</tex-math></inline-formula>-transitive or <inline-formula><tex-math id="M3">\begin{document}$ t $\end{document}</tex-math></inline-formula>-homogeneous permutation group on a base block, which yields a <inline-formula><tex-math id="M4">\begin{document}$ t $\end{document}</tex-math></inline-formula>-design in general. It is open how to use a <inline-formula><tex-math id="M5">\begin{document}$ t $\end{document}</tex-math></inline-formula>-transitive or <inline-formula><tex-math id="M6">\begin{document}$ t $\end{document}</tex-math></inline-formula>-homogeneous permutation group to construct a <inline-formula><tex-math id="M7">\begin{document}$ (t+1) $\end{document}</tex-math></inline-formula>-design in general. It is known that the general affine group <inline-formula><tex-math id="M8">\begin{document}$ {\mathrm{GA}}_1( {\mathrm{GF}}(q)) $\end{document}</tex-math></inline-formula> is doubly transitive on <inline-formula><tex-math id="M9">\begin{document}$ {\mathrm{GF}}(q) $\end{document}</tex-math></inline-formula>. The classical theorem says that the group action by <inline-formula><tex-math id="M10">\begin{document}$ {\mathrm{GA}}_1( {\mathrm{GF}}(q)) $\end{document}</tex-math></inline-formula> yields <inline-formula><tex-math id="M11">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-designs in general. The main objective of this paper is to construct <inline-formula><tex-math id="M12">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>-designs with <inline-formula><tex-math id="M13">\begin{document}$ {\mathrm{GA}}_1( {\mathrm{GF}}(q)) $\end{document}</tex-math></inline-formula> and o-polynomials. O-polynomials (equivalently, hyperovals) were used to construct only <inline-formula><tex-math id="M14">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-designs in the literature. This paper presents for the first time infinite families of <inline-formula><tex-math id="M15">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>-designs from o-polynomials (equivalently, hyperovals).</p>