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In [C. Ding, An infinite family of Steiner systems $S(2,4,2^m)$ from cycliccodes, {\em J. Combin. Des.} 26 (2018), no.3, 126--144], Ding constructed afamily of Steiner systems $S(2,4,2^m)$ for all $m \equiv 2 \pmod{4}$ from afamily of extended cyclic codes. The objective of this paper is to present afamily of Steiner systems $S(2,4,2^m)$ for all $m \equiv 0 \pmod{4}$ supportedby a family of extended cyclic codes. The main result of this paper complementsthe previous work of Ding, and the results in the two papers will show thatthere 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 parametersof other $2$-designs supported by this family of extended cyclic codes.

Steiner systems $S(2,4,2^m)$ for $m \equiv 0 \pmod{4}$ supported by a family of extended cyclic codes