Steiner systems S(2,6,121/126), S(2,7,169) based on difference families
Ivan Hetman
TL;DR
The paper tackles constructing Steiner systems $S(2,6,v)$ and $S(2,7,v)$ from difference families, with emphasis on $v=121,126,169$. It introduces a parallelizable, calculation-based algorithm that builds base blocks over $\mathbb{Z}_v$, enforces a maximization criterion for the initial blocks, and uses a mirroring scheme to generate (potentially non-isomorphic) families, followed by isomorphism filtering. Key contributions include new $S(2,6,121)$, $S(2,6,126)$, and $S(2,7,169)$ constructions, exhaustive lists for $(121,6,1)$ and $(126,6,1)$, and a complete set of $(169,7,1)$ families derived from a single base class, along with several non-existence results for line lengths $7$–$11$. The work highlights both the potential and limitations of a computational approach, noting that formal proofs of exhaustiveness are not provided but results align with existing data for smaller parameters, and it contributes practical methods for expanding the catalog of large Steiner systems.
Abstract
In this paper new Steiner systems $S(2,6,121)$, $S(2,6,126)$, $S(2,7,169)$ are introduced. Also some non-existence results for line lengths $7..11$ are presented. There is no solid proof that presented algorithm is exhaustive or correct, but it produces same results on already known difference families for line lengths $3..6$. Due to calculation-based approach this paper probably won't be published, but will be submitted to arxiv as it contains some new results