Steiner systems $ S(2,4, 2^m) $ supported by a family of extended cyclic codes

Abstract

<p style='text-indent:20px;'>In [C. Ding, An infinite family of Steiner systems <inline-formula><tex-math id="M2">\begin{document}$ S(2, 4, 2^m) $\end{document}</tex-math></inline-formula> from cyclic codes, <i>J. Combin. Des.</i> 26 (2018), no.3, 126–144], Ding constructed a family of Steiner systems <inline-formula><tex-math id="M3">\begin{document}$ S(2, 4, 2^m) $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M4">\begin{document}$ m \equiv 2 \pmod{4} \ge 6 $\end{document}</tex-math></inline-formula> from a family of extended cyclic codes. The objective of this paper is to present a family of Steiner systems <inline-formula><tex-math id="M5">\begin{document}$ S(2, 4, 2^m) $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M6">\begin{document}$ m \equiv 0 \pmod{4} \ge 4 $\end{document}</tex-math></inline-formula> supported by this family of extended cyclic codes. The main result of this paper complements the previous work of Ding, and the results in the two papers will show that there exists a binary extended cyclic code that can support a Steiner system <inline-formula><tex-math id="M7">\begin{document}$ S(2, 4, 2^m) $\end{document}</tex-math></inline-formula> for all even <inline-formula><tex-math id="M8">\begin{document}$ m \geq 4 $\end{document}</tex-math></inline-formula>. Furthermore, this paper also determines the parameters of other <inline-formula><tex-math id="M9">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-designs supported by this family of extended cyclic codes.</p>