BIB-Designs from Circular Nearrings

Abstract

Let ${(N, \Phi)}$ be a finite circular Ferrero pair. We define the disk with center b and radius ${a, \mathcal{D}(a;b)}$, as

$$\mathcal{D} (a; b) = \{x \in \Phi(r)+c \mid r \neq 0, b\in \Phi (r)+c, |(\Phi (r)+c) \cap ( \Phi(a)+b)|=1\}.$$

Using this definition we introduce the concept of interior part of a circle, ${\Phi(a)+b}$, as the set ${\mathcal{I}(\Phi (a)+b)=\mathcal{D} (a; b) \setminus (\Phi (a)+b)}$. Moreover, if ${\mathcal{B}^{\mathcal{D}}}$ is the set of all disks, then, in some interesting cases, we show that the incidence structure ${(N, \mathcal{B}^{\mathcal{D}}, \in)}$ is actually a balanced incomplete block design and we are able to calculate its parameters depending on |N| and ${|\Phi|}$.