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

<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 $\end{document}</tex-math></inline-formula>-transitive or id="M3">\begin{document}$ $\end{document}</tex-math></inline-formula>-homogeneous permutation on base block, which yields id="M4">\begin{document}$ $\end{document}</tex-math></inline-formula>-design in general. It open how use id="M5">\begin{document}$ id="M6">\begin{document}$ construct id="M7">\begin{document}$ (t+1) known that general affine id="M8">\begin{document}$ {\mathrm{GA}}_1( {\mathrm{GF}}(q)) $\end{document}</tex-math></inline-formula> doubly transitive id="M9">\begin{document}$ {\mathrm{GF}}(q) $\end{document}</tex-math></inline-formula>. The theorem says by id="M10">\begin{document}$ id="M11">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-designs main objective this paper id="M12">\begin{document}$ 3 with id="M13">\begin{document}$ and o-polynomials. O-polynomials (equivalently, hyperovals) were used only id="M14">\begin{document}$ literature. This presents for first time infinite families id="M15">\begin{document}$ from o-polynomials hyperovals).</p>