Further gain-function constructions from Steiner systems

Find other examples of gain functions on Steiner systems that yield generalized quadrangles.

Background

The general construction in the paper starts with an incidence gain graph whose underlying incidence structure is a linear space. Theorem 5 characterizes when the construction produces a generalized quadrangle: the associated rho functions must be bijective. The affine-plane example provides one concrete family satisfying this condition.

The paper leaves unresolved whether analogous gain functions exist for other Steiner systems and asks for additional examples beyond the affine-plane construction. Such examples would expand the family of generalized quadrangles obtainable through the incidence gain graph method.

References

We end with two open problems for further study.

Determine all of the gain functions on an affine plane over a field that yield a generalized quadrangle as in Theorem 7. Do two different such gain functions (on the same affine plane) yield isomorphic generalized quadrangles?

Find other examples of gain functions on Steiner systems that yield generalized quadrangles.

— Incidence Gain Graphs and Generalized Quadrangles  (2502.01805 - McCulloch, 3 Feb 2025) in Section 4, 'Affine Plane Examples and Further Inquiry,' second displayed Problem following Theorem 7