# Report on"Combinatorial constructions under group actions"

During the second year of the project "Combinatorial constructions under group actions" P12642-MAT a lot of work was devoted to the enumeration of regular matroids. Matroids are a combinatorial model for independence structures, generalizing different types of dependency relations like for instance linear dependency and algebraic dependency. In this general setting we are speaking of independent and dependent sets, of bases and rank functions of closure operators, hyperplanes and closed sets. A matroid, which can be described by linear dependency in a vector space over a field K is called K-linear. Regular matroids are K-linear for any field K. Considering certain connections between isometry classes of linear codes (which were dealt with during the first year of this project, cf. [1]) and the isomorphism classes of binary matrix-matroids, we succeeded in implementing an algorithm which checks binary matroids for regularity. These ideas go back to M. Wild [9][10]. The isomorphism classes of binary matrix matroids can be described as orbits under group actions. Orbit representatives are then checked for regularity by applying a theorem of R. E. Bixby [2]. This way we are checking whether the given matroid is a series extension of the Fano or the dual Fano matroid. By doing this we get some further information about these matroids, for instance the set of all hyperplanes of it, or the set of all closed sets (of given rank). An article collecting all these results is just in preparation.

Mathematical music theory was another research topic. (Maybe it was not described so explicitly in our research plan, but an invitation to write a longer survey article on this topic and lectures given at three conferences were changing our plans.) The article [6] dealing with the enumeration of mosaics appeared in Discrete mathematics. Moreover Harald Fripertinger was presenting these results at two scientific meetings: First at the conference Algebraic Combinatorics and Applications in Gößweinstein (Germany), which was dedicated to the 60-th birthday of Prof. A. Kerber, and then at the 7. Österreichisches Mathematikertreffen in Graz. In addition to this he gave a survey lecture on Enumeration and Construction in Music Theory at the Diderot Forum on Mathematics and Music, which took place in Vienna. A summary of this talk [5] was published in the conference proceedings. Furthermore Harald Fripertinger was invited by M. Boroda, who is the editor of "Musikometrika", to prepare an article about enumeration and construction of motives. This article [4] will appear in a special issue of Musikometrika, which will be dedicated to the concept of motives. It is shown in all details how the number of essentially different (i. e. not similar) motives can be computed and how to construct a (complete) system of representatives of motives. This paper is quite long since all necessary mathematical notions and definitions concerning sets, functions, permutations, elementary number theory, group theory, permutation groups and group actions are presented.

In coding theory we were continuing the research from last year. The article [7] about random generation of linear codes appeared in Aequationes mathematicae.

Existing programs (developed in SYMMETRICA) were rewritten and improved in order to correct bugs. These programs can be used for the construction of generator matrices of binary linear (n,k)-codes of given minimum distance. And there exists a database program for handling binary codes of optimal minimum distance. It manipulates both lower and upper bounds for the minimum distance and generator matrices of binary (n,k)-codes as it is described in the first chapter of [1]. It allows to compute various new codes from a given code by parity check columns, punctuation, shortening, A-construction, Y1-construction and B-construction, direct sum, (u,u+v)-construction or by a tensor product.

Together with the supervisor of this project an article [8] is prepared which is dealing with some functional equations in, and applications of group actions to astronomy.

The page proofs for [3] were already corrected, but the article is still not printed.

• References

• harald.fripertinger@kfunigraz.ac.at,
last changed: January 23, 2001