Classification and isomorphism of affine-plane gain functions

Determine all gain functions on an affine plane over a field that produce a generalized quadrangle via Construction \mathfrak{M} as in Theorem 7, and determine whether two distinct such gain functions on the same affine plane produce isomorphic generalized quadrangles.

Background

The paper constructs a gain function on the incidence graph of an affine plane over an arbitrary field and proves that the resulting incidence structure is a generalized quadrangle. For finite fields of order q, the construction produces a generalized quadrangle denoted GQ(q+1,q-1), which computational checks indicate is isomorphic to the dual of the Payne generalized quadrangle P(W(q),x) for q up to 16.

The problem asks for a complete classification of all gain functions on an affine plane that satisfy the bijectivity condition required by Theorem 5, as well as a determination of whether different qualifying gain functions can yield nonisomorphic or necessarily isomorphic generalized quadrangles. Both questions are explicitly left unresolved.