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

In [C. Ding, An infinite family of Steiner systems $S(2,4,2^m)$ from cyclic codes, {\em J. Combin. Des.} 26 (2018), no.3, 126–144], Ding constructed a family of Steiner systems $S(2,4,2^m)$ for all $m\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}4)$ from a family of extended cyclic codes. The objective of this paper is to present a family of Steiner systems $S(2,4,2^m)$ for all $m\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$ supported by a 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 $S(2,4,2^m)$ for all even $m\ge 4$. This paper also determines the parameters of other $2$-designs supported by this family of extended cyclic codes.