Infinite families of 2-designs from two classes of binary cyclic codes with three nonzeros

Abstract

<p style='text-indent:20px;'>Combinatorial <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula>-designs have been an interesting topic in combinatorics for decades. It is a basic fact that the codewords of a fixed weight in a code may hold a <inline-formula><tex-math id="M2">\begin{document}$ t $\end{document}</tex-math></inline-formula>-design. Till now only a small amount of work on constructing <inline-formula><tex-math id="M3">\begin{document}$ t $\end{document}</tex-math></inline-formula>-designs from codes has been done. In this paper, we determine the weight distributions of two classes of cyclic codes: one related to the triple-error correcting binary BCH codes, and the other related to the cyclic codes with parameters satisfying the generalized Kasami case, respectively. We then obtain infinite families of <inline-formula><tex-math id="M4">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-designs from these codes by proving that they are both affine-invariant codes, and explicitly determine their parameters. In particular, the codes derived from the dual of binary BCH codes hold five <inline-formula><tex-math id="M5">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>-designs when <inline-formula><tex-math id="M6">\begin{document}$ m = 4 $\end{document}</tex-math></inline-formula>.</p>