BIB-Designs from Circular Nearrings |
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Authors: | Anna Benini Achille Frigeri Fiorenza Morini |
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Institution: | 1. DICATAM Dip. Ing. Civ. Arch. Terr. Amb. e Matematica, Università degli Studi di Brescia, via Valotti, 9, 25133, Brescia, Italy 2. Dipartimento di Elettronica e Informazione, Politecnico di Milano, Via Ponzio, 34/5, 20133, Milano, Italy 3. Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze, 53/A, 43124, Parma, Italy
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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 ${|\Phi|}$ . |
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