Classification of affine-plane gain functions yielding generalized quadrangles

Determine all gain functions on an affine plane over a field for which Construction \(\mathfrak{M}\) yields a generalized quadrangle as in Theorem 7.

Background

Theorem 7 gives one explicit gain function on the incidence graph of an affine plane over a field and proves that the associated Construction M\mathfrak{M} produces a generalized quadrangle. The construction depends on gain functions satisfying the bijectivity condition for the associated rho functions.

The paper leaves unresolved the classification problem of identifying every gain function on an affine plane that satisfies the required condition and consequently yields a generalized quadrangle.

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.

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

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?

— Incidence Gain Graphs and Generalized Quadrangles  (2502.01805 - McCulloch, 3 Feb 2025) in Section 2, “Affine Plane Examples and Further Inquiry,” first Problem environment