Isomorphism classification of generalized quadrangles from affine-plane gain functions

Determine whether two distinct gain functions on the same affine plane that yield generalized quadrangles through Construction \(\mathfrak{M}\) necessarily produce isomorphic generalized quadrangles, and, if not, classify the resulting isomorphism types.

Background

Different gain functions can be placed on the same affine-plane incidence graph, and the construction is sensitive to the gain data subject to the rho-function bijectivity criterion. The paper establishes that switching gain functions induces isomorphic incidence structures, but it does not determine whether all admissible, possibly nonswitching-equivalent gain functions produce the same generalized quadrangle up to isomorphism.

The first displayed open problem explicitly asks whether two such gain functions on one affine plane yield isomorphic generalized quadrangles. Resolving this would clarify the extent to which the construction produces genuinely distinct generalized quadrangles.