New Steiner systems $S(2,6,v)$ with block length 6

TL;DR

This paper catalogs various Steiner systems with block size 6, discovering new types and specific instances, including 1-rotational designs for certain groups and new systems with specific parameters.

Contribution

It introduces previously unknown 1-rotational Steiner systems for specific groups and lists new Steiner systems with parameters 96, 106, and 111.

Findings

Discovery of 1-rotational designs for groups SL(2,5) and a semidirect product group.

Enumeration of new Steiner systems S(2,6,96), S(2,6,106), S(2,6,111).

Expansion of known Steiner systems with explicit constructions.

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.