An infinite family of Steiner systems $S(2,4,2^m)$ from cyclic codes

Steiner systems are a fascinating topic of combinatorics. The most studied Steiner systems are $S(2,3,v)$ (Steiner triple systems), $S(3,4,v)$ (Steiner quadruple systems), and $S(2,4,v)$. There are a few infinite families of Steiner systems $S(2,4,v)$ in the literature. The objective of this paper is to present an infinite family of Steiner systems $S(2,4,2^m)$ for all $m\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}4)\ge 6$ from cyclic codes. This may be the first coding-theoretic construction of an infinite family of Steiner systems $S(2, 4, v)$\mathrm{.}Asaby-product,manyinfinitefamiliesof$2$-designs are also reported in this paper.