In this project (a joint project with the Taiwanese National Science Council), researchers from Tainan (Taiwan), Hamburg, and Linz will work based on their previous research in the following relevant areas:

**Construction of Frobenius groups:**For applications like building designs with specific parameters, it is necessary to provide Frobenius groups of specified size and isomorphism type (cf. [Boy01]). [KK95] presents a characterization of Frobenius groups where both the kernel and the complement are abelian. In general, Frobenius groups with abelian kernel can be obtained via representations of fixed-point-free automorphism groups over finite fields [May98,May00]. These representations can be found using [Bro01].For the construction of Frobenius groups with non-abelian kernel, there is no systematic approach known. Examples of such groups have been described, e.g., in [KW91,Aic01].

**Designs from planar near-rings:**Various ways to obtain designs and even BIB-designs from planar near-rings (or equivalently from Frobenius groups) are presented in [Cla92]. We give one example: Let be a planar near-ring, take the elements of as points and take the set of with as blocks. Then the design with these points and blocks is a BIB-design.Some near-rings (Frobenius groups) yield circular designs, that is, designs where any two blocks intersect in at most two points. In this case, the corresponding near-rings (Frobenius groups) are called circular as well. The combinatorial structure of circular designs has been investigated in [Ke92,KK96]. As an example of the interplay between design and group structure, it is shown in [BFK96] that only Frobenius groups with metacyclic complement give circular designs. Moreover, in that paper it is shown that if is circular for a -group , then all Frobenius groups with a -group (same , same ) are circular. The question remains open whether for any metacyclic group which admits a fixed-point-free action, there exists a group such that yields a circular design.

In [Pil96] applications of BIB-designs for statistical purposes (experimental design and analysis) are presented.

**Codes from planar near-rings:**Certain binary codes associated with (circular) designs coming from planar near-rings (Frobenius groups) reach the Johnson bound. These codes are optimal in the sense that they realize the maximal possible number of codewords with fixed weight, length and minimum distance [FHP90,Fuc91].**Structure theory of near-rings:**In the theory of near-rings, fixed-point-free automorphism groups come into play at two different places.- All planar near-rings can be constructed via Ferrero's method [Fer70].
- Every zero-symmetric -primitive near-ring with unit is either a ring, or it is dense in a centralizer near-ring , where is a fixed-point-free automorphism group on . The ideas used in the proof of this result have been adapted to -primitive near-rings without unit in [FP89].

- Structural characterizations of planarity: For example, a finite zero-symmetric near-ring is integral and planar iff it is subdirectly irreducible and there is a natural number such that for all .
- Geometrical structures: Planar near-rings give rise to geometrical structures, namely BIB-designs. It might be that similar structures can be obtained from other classes of -primitive composition algebras.

**Functions on Frobenius groups:**In [Aic01] a formula for the number of polynomial functions on certain semi-direct products of groups is given. According to this result the number of polynomial functions on a Frobenius group ( normal) is equal to**Computing with near-rings:**A computer algebra package for near-rings (``SONATA")[ABE$^+$00] based on GAP was developed in Linz (FWF-Projects P11486-TEC and P12911-INF). This allows us to study planar near-rings and designs on a computer. SONATA has already established itself as an important tool in near-ring theory.