Enumerations of 1-rotational Steiner systems
Ivan Hetman, Taras Banakh, Alex Ravsky
TL;DR
The work addresses enumerating 1-rotational Steiner systems $S(2,k,v)$ across admissible parameters by extending cyclic/difference-family methods to general group actions. It presents a generalized algorithmic framework and uses GAP LOOPS to generate group actions, coupled with a difference-family equivalence to filter nonessential duplicates. The authors provide extensive enumerations for $k=3,4,5$ and an in-depth study of 1-rotational unitals of order $4$, including counts of non-isomorphic designs across various automorphism groups and explicit examples. This yields a substantial expanded catalog of 1-rotational Steiner systems, offering a resource for theoretical insights and future computational explorations in combinatorial design theory.
Abstract
In this paper new $1$-rotational 2-Steiner systems for different admissible $v,k$ pairs are introduced. In particular, $1$-rotational unitals of order $4$ are enumerated.