Circular planar nearrings: geometrical and combinatorial aspects

Let $(N,Φ)$ be a circular Ferrero pair. We define the disk with center $b$ and radius $a$, $\mathcal{D}(a;b)$, as $\mathcal{D}(a;b)=\{x\in Φ(r)+c\mid r\neq 0,\ b\in Φ(r)+c,\ |(Φ(r)+c)\cap (Φ(a)+b)|=1\}.$ We prove that in the field-generated case there are many analogies with the Euclidean geometry. 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.

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