Existence of Steiner designs with the proposed tight-dual-arc parameters

Establish or rule out the existence of Steiner 2-designs with the parameter families given by equations (eqseries), in particular the cases with s≥3 and the listed parameter sets, together with the corresponding maximal arcs and tight dual arcs when required.

Background

The paper derives a parametric family of Steiner 2-design parameters supporting a maximal arc whose dual is a tight dual arc. Such a tight dual arc would produce a non-canonical maximum clique in the block graph, potentially yielding infinitely many examples relevant to the unresolved infinite-family problem.

Beyond the known smallest non-affine case, namely the 2-(66,6,1) designs corresponding to s=3 and t=2, the existence of designs in this parameter family is unknown. The paper specifically identifies several next cases and notes that no construction or nonexistence proof is currently available.

References

Existence remains an open question for other values of $s\ge 3$ and~$t$, but this series of parameters could potentially hold an infinite family of examples for Problem~\ref{prob:NonCanonicalFamiliesWithoutDesignStructure}.

— Erdős-Ko-Rado properties of Steiner 2-designs  (2609.26607 - Adriaensen et al., 22 Sep 2026) in Section 3, immediately following the list of the cases (s,t)=(3,3),(4,2),(5,2),(4,3)