Steiner systems $S(2,6,226)$ and $S(2,6,441)$ do exist!
Taras Banakh, Ivan Hetman, Alex Ravsky
TL;DR
This work resolves two previously undecided cases in the existence of Steiner systems $S(2,6,v)$ by constructing seven $S(2,6,226)$ and six $S(2,6,441)$ through computer-assisted searches using $1$-rotational difference families in specific finite groups. The methods extend to one-point extensions with an extra point $\infty$, and the resulting designs are shown to be pairwise non-isomorphic via a fingerprint invariant. The approach demonstrates concrete, scalable techniques for generating large Steiner systems and expands the catalog of known $S(2,6,v)$ designs, contributing to the understanding of existence conditions in combinatorial design theory.
Abstract
A Steiner system $S(2,k,v)$ is a set $X$ of cardinality $v$ endowed with a family $\mathcal L$ of $k$-element subsets of $X$ such that any two distinct points of $X$ belong to a unique set of the family $\mathcal L$. If a Steiner system $S(2,k,v)$ exists, then $k-1$ divides $v-1$ and $k(k-1)$ divides $v(v-1)$. Those divisibility conditions are necessary but not sufficient for the existence of a Steiner system $S(2,k,v)$. For instance, the Bruck--Ryser Theorem implies that no Steiner system $S(2,6,36)$ exists, despite $5$ divides $35$ and $6\cdot 5$ divides $36\cdot 35$. On the other hand, Wilson showed that for every natural number $k\ge 2$, a Steiner system $S(2,k,v)$ exists for all but finitely many natural numbers $v$ satisfying the above divisibility conditions. The Handbook of Combinatorial Designs lists 29 numbers $v$ for which the existence of a Steiner system $S(2,6,v)$ is not known: 51, 61, 81, 166, 226, 231, 256, 261, 286, 316, 321, 346, 351, 376, 406, 411, 436, 441, 471, 501, 561, 591, 616, 646, 651, 676, 771, 796, 801. In this paper we present seven Steiner systems $S(2,6,226)$ and six Steiner systems $S(2,6,441)$ thus resolving two of those 29 undecided cases. The discovered Steiner systems $S(2,6,226)$ and $S(2,6,441)$ were found by computer search, as $1$-rotational difference families and difference families for the groups $(\mathbb Z_5\times\mathbb Z_5\times\mathbb Z_3)\rtimes\mathbb Z_3$ and $(\mathbb Z_7\rtimes\mathbb Z_3)\times(\mathbb Z_7\rtimes\mathbb Z_3)$, respectively.