A class of functions and their application in constructing semi-biplanes and association schemes

We give an alternative proof of the fact that a planar function cannot exist on groups of even order. The argument involved leads us to define a class of functions which we call semi-planar. Through the introduction of an incidence structure we construct semi-biplanes using semi-planar functions. The method involved represents a new approach to constructing semi-biplanes and provides infinite classes of semi-biplanes unlike any known to the authors. For a particular class of semi-planar functions, we provide a method to construct association schemes with two associate classes. Such an association scheme is equivalent to a strongly regular graph.