Existence of $S(2,9,369)$, new unitals of order $6$ and other Steiner systems with block length $\ge 7$
Ivan Hetman
TL;DR
The paper tackles the scarcity of $S(2,k,v)$ examples for $k\ge 7$ by developing a subgroup-action construction on a highly symmetric design to generate new Steiner systems via orbit decompositions. It reports the existence of $S(2,9,369)$ and several new unitals and designs, including $S(2,7,217)$, $S(2,7,175)$, $S(2,7,259)$, $S(2,8,120)$, $S(2,8,504)$, and $S(2,9,513)$, with block data encoded by fingerprints and publicly available. The construction relies on selecting a subgroup $G$ of the automorphism group to yield an explicit orbit partition and searching for basic blocks; data are provided with fingerprints and are accessible at a Git repository. The authors propose two infinite-series conjectures: one with $G=PSL(2, F_k)$ yielding $S(2,k,|G|+k)$ and one with $G=SL(2, F_k)$ yielding $S(2,k+1,k^3+1)$, with partial confirmations up to $k\le 9$ and connections to Hall plane unitals for small $k$. These results advance the constructive theory of Steiner systems and hint at broad families of designs.
Abstract
Whereas Steiner systems $S(2,k,v)$ with block length $k \le 5$ have large amount of examples and the existence is established for all admissible $v$, for $k\ge 6$ only few examples are known even for decided cases. In this paper the existence of $S(2,9,369)$ is established and some new examples for other admissible pairs $(k,v)$ are given. In particular, lots of new unitals of order $6$ (or $S(2,7,217)$) together with $S(2,7,175)$, $S(2,7,259)$, $S(2,8,120)$, $S(2,8,504)$, $S(2,9,513)$ are presented. Found examples suggest two conjectures on infinite series of designs.