Fiorenza Morini

A problem on partial sums in abelian groups

In this paper we propose a conjecture concerning partial sums of an arbitrary finite subset of an abelian group, that naturally arises investigating simple Heffter systems. Then, we show its connection with related open problems and we present some results about the validity of these conjectures.

Codes and combinatorial structures from circular planar nearrings

Nearrings are generalized rings in which addition is not in general abelian and only one distributive law holds. Some interesting combinatorial structures, as tactical configurations and balanced incomplete block designs (BIBDs) naturally arise when considering the class of planar and circular nearrings. In [12] the authors define the concept of disk and prove that in the case of field-generated planar circular nearrings it yields a BIBD, called disk-design. In this paper we present a method for the construction of an association scheme which makes the disk-design, in some interesting cases, an union of partially incomplete block designs (PBIBDs). Such designs can be used in the construction of some classes of codes for which we are able to calculate the parameters and to prove that in some cases they are also cyclic.

Partially balanced incomplete block designs from weakly divisible nearrings

In [[6] Riv. Mat. Univ. Parma 11 (2) (1970) 79-96] Ferrero demonstrates a connection between a restricted class of planar nearrings and balanced incomplete block designs. In this paper, bearing in mind the links between planar nearrings and weakly divisible nearrings (wd-nearrings), first we show the construction of a family of partially balanced incomplete block designs from a special class of wd-nearrings; consequently, we are able to give some formulas for calculating the design parameters.

Composition rings from formal power series rings

ABSTRACT We study the ideals in some Dickson nearrings of polynomials and formal power series. For some of their related quotients, we introduce variants and generalizations, and construct composition rings also.

New designs from circular nearrings

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

A study of some Dickson nearrings

\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\}.

Circular planar nearrings: geometrical and combinatorial aspects

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 \({|\Phi|}\).