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

Abstract

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 $m\equiv 2(\mathrm{mod}4)\ge 6$ from cyclic codes. This may be the first coding-theoretic construction of an infinite family of Steiner systems $S(2,4,v)$. As a by-product, many infinite families of $2$-designs are also reported in this paper.

arXiv admin note: text overlap with arXiv:1605.03796

References in corpus (2)

• Infinite families of 2-designs and 3-designs from linear codes
• Another Generalization of the Reed-Muller Codes