Existence of $S(2,9,369)$, new unitals of order $6$ and other Steiner systems with block length $\ge 7$

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.