New Steiner systems $S(2,6,v)$ with block length 6
Taras Banakh, Ivan Hetman, Alex Ravsky
TL;DR
This work advances the enumeration of Steiner systems $S(2,6,v)$ by employing a mixed algorithm that leverages generalized and commutative-group techniques to construct and classify $1$-rotational designs for $k=6$. It reports new $1$-rotational designs for $SL(2,5)$ and $( obreak\mathbb Z_3 \times \nobreak\mathbb Z_3) \rtimes \nobreak\mathbb Z_3 \;\times\; \nobreak\mathbb Z_5$, and introduces several novel families, including $S(2,6,66)$ with automorphism group of order $20$, $S(2,6,111)$, and a Denniston-type $S(2,6,96)$ yielding a large collection of designs. The paper further explores designs generated by cyclic and semidirect-product groups on various orbit decompositions, producing dozens to hundreds of nonisomorphic designs (e.g., 4 designs for $v=111$ under $\mathbb Z_{55}$, 513 designs for a three-9orbit construction with $\mathbb Z_{37}\rtimes \mathbb Z_3$, and 634 designs for $S(2,6,96)$ under $\mathbb Z_{19}\rtimes \mathbb Z_3$). The results enrich the catalog of $S(2,6,v)$ and illustrate rich group-action structures that yield many nonisomorphic blocks, with explicit block lists and online resources for further exploration. These findings have implications for the design of combinatorial structures and their automorphism groups in finite geometry and related applications.
Abstract
In this paper various Steiner systems $S(2,k,v)$ for $k = 6$ are collected and enumerated for specific constructions. In particular, two earlier unknown types of $1$-rotational designs are found for the groups $SL(2,5)$ and $((\mathbb Z_3 \times \mathbb Z_3) \rtimes \mathbb Z_3) \times \mathbb Z_5$. Also new Steiner systems $S(2,6,96), S(2,6,106), S(2,6,111)$ are listed.