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 $m \equiv 2 \pmod{4}$ 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 $m \equiv 0 \pmod{4}$ 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 $m \geq 4$. This paper also determines the parameters of other $2$-designs supported by this family of extended cyclic codes.