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An infinite family of Steiner systems $S(2, 4, 2^m)$ from cyclic codes (1701.05965v1)
Published 21 Jan 2017 in cs.IT, math.CO, and math.IT
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 $S(2, 4, 2m)$ for all $m \equiv 2 \pmod{4} \geq 6$ 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.