Existence of infinitely many biplanes

Construct infinitely many symmetric 2-designs with parameters (v, d, 2), equivalently biplanes of order d - 2, for which the associated incidence graphs are d-regular bipartite graphs whose median eigenvalues are exactly plus or minus sqrt(d - 2).

Background

The paper observes that the incidence graph of a symmetric 2-design with parameters (v, d, 2), also called a biplane of order d - 2, would yield a d-regular bipartite graph with median eigenvalues exactly plus or minus sqrt(d - 2), where v = d(d - 1)/2 + 1.

Thus, the existence of infinitely many such biplanes would provide an infinite family attaining the relevant median-eigenvalue bound. The authors identify this as a long-standing unresolved problem in design theory and note that computational searches found no biplanes of orders between 8 and 5000.