Existence of the first further tight-dual-arc design cases

Determine whether Steiner 2-designs exist for the specific parameter sets 2-(153,9,1), 2-(176,8,1), 2-(370,10,1), and 2-(408,12,1), with the associated maximal and tight dual arcs described by the parameter family when applicable.

Background

The paper lists four subsequent parameter cases arising from the proposed tight-dual-arc construction. The first three require designs with parameters 2-(153,9,1), 2-(176,8,1), and 2-(370,10,1), while the fourth involves 2-(408,12,1) and a related 2-(297,9,1) design.

The authors report that they have neither constructed nor ruled out these designs. Thus, these cases constitute concrete unresolved existence questions, and resolving them could produce further non-canonical maximum cliques and potentially contribute to an infinite family.

References

We have not succeeded to construct or rule out any of these designs. Indeed, in the first three cases the existence of a design with parameters as~$$ is an open question, with or without maximal arcs. In the fourth case, both $2$-$(408,12,1)$ and $2$-$(297,9,1)$ designs are unknown.

— Erdős-Ko-Rado properties of Steiner 2-designs  (2609.26607 - Adriaensen et al., 22 Sep 2026) in Section 3, final paragraph before the conclusion of the section