A class of functions and their application in constructing semi-biplanes and association schemes
The authors introduce a modification of the notion of a planar function (introduced by Dembowski and the reviewer) as follows: Let \(G\) and \(H\) be finite groups the same even order written additively but not necessarily abelian. A function \(f:G\to H\) is a semi-planar function if for every \(a\neq 0\) the equation \(f(x+a)- f(x)=y\) \((y\in H)\) has either 0 or 2 solutions. If \(f\) is a semi-planar function we can introduce an incidence structure where the point \((x,y)\) is incident with the line \({\mathcal L}(a,e)\) iff \(y=f (x-a)+e\). Here of course \(x,a\in G\) and \(y,e\in H\). If \(G\) and \(H\) are of order \(k\), the incidence structure defined by \(f\) has \(k^2\) lines, each line contains \(k\) points and each point is on \(k\) lines. The structure is self dual and each pair of points occurs in 0 lines or in 2 lines. A ``connected'' incidence structure is called a ``semi-biplane'' if (1) Each pair of points is incident either with two or with no blocks and (2) Each pair of blocks has 0 or 2 points in common. It follows that the incidence structure defined by a semi-planar function either is a semi-biplane or splits into two semi-biplanes. The authors give several examples of semi-planar functions. They also introduce ``association scheme with two associate classes''.
scientific article; zbMATH DE number 1334516