Coverings have applications in communications and computer science via error trapping decoding and data compression, respectively, and in coding theory. Recently, coverings have been used in cryptography. I am interested in constructing coverings that improve the known upper bounds on the corresponding covering numbers. I will search for various types of designs with certain extremal properties that can be used to produce other combinatorial structures; coverings in particular. These include partially balanced designs, group divisible designs and designs with maximally different blocks. I would like to explore some interconnections between design theory and other areas of mathematics, for example, algebra, graph theory, coding theory and finite geometries.
I am planning to continue the search for good coverings (coverings with the
smallest possible size). I believe that some other general constructions can be
obtained once we have good knowledge on the properties and structure of small
coverings. There is a possibility of applying
GAP (Groups Applications Package) of RWTH-Aachen and
Discreta of Betten, Laue and Wassermann in the searches.
The first two
packages can be used to generate appropriate groups. It will be generally
difficult to improve the covering
bounds in the current range of applications. The bounds have been recently
attacked by
several computer searches via optimization
acting on the set of all 
-subsets. One possible alternative that
I want to explore is experimenting with optimization algorithms
acting on the set of representatives of orbits on the -sets under the action
of a prescribed group of automorphisms of the covering. The success
of a computer search for minimal coverings would allow us to look
at the structure of these coverings and possibly find other general
direct combinatorial constructions.
Many of the known construction of combinatorial objects are based on properties of geometry structures. I am particularly interested in finding new applications of finite geometry in the constructions of coverings, packings and designs.
Recently I have developed a powerful optimization method which I want to try in various searches for designs and other combinatorial structures. I would also want to look at the possibility of using the idea of the method in continuous optimization.