Point-transitive Steiner systems S(2,6,111/121/126), S(2,7,169/175)
Ivan Hetman
TL;DR
This paper extends the search for point‑transitive Steiner systems of the forms $S(2,6,v)$ and $S(2,7,v)$ by generalizing cyclic difference families to commutative groups and exploring a non‑commutative case, introducing a hyperbolic‑frequency fingerprint to distinguish designs. It provides an extensive, data‑driven catalog of multiplier–fingerprint–difference‑family constructions across various abelian groups, highlighting known results (bolded with citations) and nonexistence outcomes. A key result is the enumeration of $S(2,6,111)$ designs with automorphism group $\mathbb{Z}_{37}\rtimes\mathbb{Z}_3$ (order $111$), asserting 30 non‑isomorphic designs and including Mills’ original construction within a comprehensive data table. The work thus broadens the landscape of point‑transitive Steiner designs and sets the stage for deeper algorithmic development and broader non‑commutative explorations in future papers.
Abstract
In this paper new Steiner systems $S(2,6,111)$, $S(2,6,121)$, $S(2,6,126)$, $S(2,7,169)$, $S(2,7,175)$ and possibly others with point-transitive (commutative except $S(2,6,111)$ case) automorphism groups are introduced.