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>

On the Intersection Distribution of Degree Three Polynomials and Related Topics

“…We mention a recent work due to Ding and Tang [10], in which polynomials over finite fields were employed to construct combinatorial t-designs. While determining the parameters of the t-design arising from a polynomial f is difficult in general [10], we note that the multiplicity distribution of f implies the parameters of the associated t-design. Therefore, this design-theoretic application supplies one more motivation to study the multiplicity distribution of polynomials over finite fields.…”

“…6),(3,5) 5(1,4),(2,10),(3,8),(4,7) 7(1,6),(2,21),(3,16),(4,15),(5,18),(6,11) 8 (1, 7), ({2, 4}, 28), ({3, 5}, 21),(6,28),(7,13) 9(1,8),(2, 36),(3,24), (4, 30),(5,24),(6,28) ,(7, 32),(8,15) 11(1,10),(2, 55), ({3, 7}, 40) , (4, 45) ,(5, 38),(6, 35),(8, 45) , (9, 50),(10,19) 13(1,12), (2, 78), (3, 56) , (4, 57) , (5, 60) , (6, 58), (7, 48), (8, 69) , (9, 56) , (10, 54) , (11, 72), (12, 23) 16 (1, 15), ({2, 8}, 120), (3, 85), (4, 60), (5, 102), (6, ...…”