"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

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.

harald.fripertinger@kfunigraz.ac.at,

last changed: January 23, 2001