Existence of infinitely many biplanes

Determine whether infinitely many biplanes exist, including whether a biplane with parameters (121,16,2) exists.

Background

The paper explains that the existence of a biplane of order q would yield a non-trivial 2-intersecting hypergraph with order approximately 2k3/27 for the corresponding parameter choice k = 3(q+2)/2.

The authors identify both the existence of infinitely many biplanes and the particular biplane with parameters (121,16,2) as unresolved design-theoretic questions.

References

However, the existence of infinitely many biplanes is still open. Cameron notes that deciding whether or not infinitely many biplanes exist, seems far out of reach. In fact, the existence of the biplane with parameters $(121,16,2)$ remains unresolved.

— On the order of intersecting hypergraphs  (2504.05162 - Cambie et al., 7 Apr 2025) in Concluding remarks, Section 5