Infinite families of 2-designs from a class of non-binary Kasami cyclic codes

Abstract

<p style='text-indent:20px;'>Combinatorial <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula>-designs have been an important research subject for many years, as they have wide applications in coding theory, cryptography, communications and statistics. The interplay between coding theory and <inline-formula><tex-math id="M2">\begin{document}$ t $\end{document}</tex-math></inline-formula>-designs has been attracted a lot of attention for both directions. It is well known that a linear code over any finite field can be derived from the incidence matrix of a <inline-formula><tex-math id="M3">\begin{document}$ t $\end{document}</tex-math></inline-formula>-design, meanwhile, that the supports of all codewords with a fixed weight in a code also may hold a <inline-formula><tex-math id="M4">\begin{document}$ t $\end{document}</tex-math></inline-formula>-design. In this paper, by determining the weight distribution of a class of linear codes derived from non-binary Kasami cyclic codes, we obtain infinite families of <inline-formula><tex-math id="M5">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-designs from the supports of all codewords with a fixed weight in these codes, and calculate their parameters explicitly.</p>