Classification of gain functions on affine planes yielding generalized quadrangles

Determine all gain functions on an affine plane over a field that yield a generalized quadrangle through the construction of Theorem 7, and determine whether two distinct such gain functions on the same affine plane yield isomorphic generalized quadrangles.

Background

Theorem 7 gives one explicit gain function on the incidence graph of an affine plane over a field and proves that the associated incidence structure is a generalized quadrangle. The paper does not classify the gain functions with this property.

The stated problem has two parts: classifying every suitable gain function and determining whether distinct suitable gain functions defined on the same affine plane can produce isomorphic generalized quadrangles.