RESEARCH EXPERIENCE

My previous experience involves working in various educational and research institutions including five universities, the three mentioned on the first page of this document, the University of Northern British Columbia (my current working place), and the Technical University of Gabrovo, Bulgaria where I worked from 1989 to 1993. I entered the Technical University with the rank of Assistant Professor after winning an academic competition. While I was working there I was involved in a group called the Young Researchers Group. Our responsibilities were, among other things, to contribute to the development of a new series of textbooks called Mathematics for Engineering Education. I am a coauthor of two of these books. During my stay at the Technical University of Gabrovo I became interested in Design Theory. Shortly after that, in 1993, I came to Canada to continue my studies at the University of Victoria, and later at Simon Fraser University. Currently, I am an Assistant Professor at the University of Northern British Columbia.

My recent research is reflected in the articles listed in the publications section, as well as in my M.Sc. and Ph.D. theses; noting that most of the results of my M.Sc and Ph.D. theses have been published in refereed journals. During my work experience I have collaborated and exchanged information with many other researchers, mostly via Internet or conference activities. I spent a year as a postdoctoral research fellow with professor K. Heinrich at Simon Fraser University. At that time, I was coordinating the meetings of the Discrete Mathematics Seminar, communicating with other researchers and giving talks myself.

I will briefly discuss some of the most significant results of my research. This discussion includes a description of work in both Ph.D. and M.Sc. theses as well as some important results beyond the doctoral dissertation. First, let me introduce some basic terms and notation. Let $D$ = $\{ B_1, B_2,..., B_b \}$ be a finite family of $k$-subsets (called blocks of a $v$-set $X(v)$ = $\{ 1, 2,..., v \}$ (with elements called points. Then $D$ is a $t$- $(v,k,\lambda )$ design if every $t$-subset of $X(v)$ is contained in exactly $\lambda$ blocks of $D$. If every $t$-subset of $X(v)$ is contained in at least $\lambda$ blocks of $D$, then $D$ is a $t$- $(v,k,\lambda )$ covering design (or covering. The number of blocks $b$ is the size of the covering, and the minimum size of a $t$- $(v,k,\lambda )$ covering is called the covering number, denoted $C_{\lambda }(v,k,t)$. A covering of size $C_{\lambda }(v,k,t)$ is called a minimal covering. Let $D$ be a $t$-$(v,k,\lambda )$ design and $p =\max_{1\leq i<j\leq b} \vert B_i\cap B_j\vert.$ Let $p^*=min\; p$, where the minimum is taken over all designs with the same parameters. Designs for which $p=p^*$ are called designs with maximally different blocks (DMDB's).

The paper  [6] is concerned with new constructions of coverings. There is no general theory behind obtaining good coverings. The existing computer algorithms produce coverings of poor quality in a reasonable amount of time or coverings of good quality but at a very large cost (CPU time). Coverings with $v \leq 32$ can be directly applied, for example, in error-trapping decoding. Since the complexity of the decoding procedure depends on the size of the covering we are interested in finding coverings of minimum size or if that is not possible, coverings of size as small as possible. There are important applications of coverings in computer science-data compression and file organization schemes. Producers of lottery software are also interested in coverings. Recently, coverings have been applied in cryptography. An interesting ``competition" started in 1993 when Nurmela and Östergård published good coverings for $v \leq 13$ obtained by simulated annealing. This method also produces some good results for $v \geq 14$, but is quite time consuming. In 1995, Gordon, Kuperberg and Patashnik tabulated known results and expanded the tables to $v \leq 32$ and $t\leq 8$ using less precise, but faster algorithms. The same year Chang, Etzion and Wei published a paper in which they improve some of the results in Gordon et al. by using combinatorial constructions based on previous results of Etzion. In  [6] we improve many of the bounds in those papers. The significance of our paper is that it provides many constructions for coverings which have smaller size than those currently being obtainable by computer methods. Most of the improvements are achieved by purely combinatorial arguments, while others are assisted by computer searches. In the process of writing the final draft of this paper I contacted other researchers asking if they could suggest any improvements, similar results or alternative solutions. H. Hämäläinen responded with additional results and he became a coauthor. The paper was written entirely by me, according to suggestions from H. Hämäläinen, K. Heinrich, T. Etzion, and the referees.

In  [5] I found three new families of minimal $(t+1)$-coverings obtained from $t$-designs. The coverings produce new covering numbers for an infinite number of parameters. Aside from $t$-designs (which are also t-coverings) prior to this work only three infinite families of coverings with $t\geq 3$ were known, those obtained by Ray-Chaudhuri in 1968, Abraham, Ghosh and Ray-Chaudhuri in 1968, and Todorov in 1984.

Designs are used in statistical planning of experiments. In  [1], an extension theorem for $t$-designs is proved. As an application, a family of $4$-$(4^m+1,5,2)$ designs is constructed by extending designs related to the 3-designs formed by the minimum weight vectors in the Preparata code of length $n=4^m, \ m\geq 2$. At that time, the only known construction of a $3$-$(16,6,4)$ design with $p=3$ was the one obtained from the Preparata codes. I found an alternative solution and used it to construct the smallest representative of the new family, the $4$-$(17,5,2)$ design $(m=2)$. I sent this construction to Tonchev at Michigan Technological University who with Baartmans generalized it. This was the first example of an infinite family of designs with $t \geq 4$ and constant $\lambda$. Recently, Bierbrauer found other examples, but our family is still the only example of an infinite family of designs with $t \geq 4$ obtained from codes. It is also the only family of designs with the smallest $\lambda$ possible for a design with repeated blocks. Note that a $4$-$(4^m+1,5,2)$ design without repeated blocks is not known to exist even in the smallest case $m=2$.

In  [2] we introduce a new method for finding designs and use it to study the existence of designs with maximally different blocks. The paper  [3] is a continuation of  [2]. In addition to the method described in  [3] we employ variations on the greedy algorithm for finding designs and partitions of a $t$-design into $(t-1)$-designs. In  [4] we investigate the intersection number of known designs. We prove that designs obtained from codes in works of Assmus and Matson, and MacWilliams, Odlyzko and Sloan are designs with small intersection number. We then use a result of Driessen and a previously unknown corollary to this result to prove the existence of over 500 new 3,4, and 5-designs. In  [7] we combine methods from  [4] and  [3] with the proof that a design obtained from a code of van Lint and MacWilliams is a design with small intersection number to prove the existence of new 3-designs. Part of the designs are obtained from the inversive geometry of order 5 via a method described in  [3] and  [2].

In a recent joint paper with M. Greig and K. Heinrich  [11] we find new infinite classes of covering numbers. Some of the classes depend on two or three different parameters, so any particular value of one of the parameters produces a separate infinite class thus resolving the existence of minimal coverings for quite a wide spectrum of parameters. Postscript files of recent papers can be found at my web-site (see the beginning of this document). In March, 1998 I was invited at the University of Nebraska by Spyros Magliveras and Douglas Stinson to give talks on coverings, and to work on large sets of designs. This visit resulted in the joint paper  [10]. I have also colaborated with K. Heinrich on two other papers,  [9] and  [8]. In  [8] we found some further generalizations of the results from  [6]. The paper  [9] complements the works of other researchers on super-simple designs.

Recently I became interested in combinatorial optimization and the possibilities of using combinatorial optimization as a computational tool for solving problems in various areas of discrete mathematics. Some of the results in  [6],  [12] and  [13] were found by optimization. These works gave me the opportunity to study the existing optimization methods, compare their performance, notice some common weaknesses of these methods and eventually develop a new method that overperforms (and generalizes) all of the existing methods. I call it a Level Dependent Experimental Optimization. The potential of the new method has been exploited, among other tools, in several new works (still in progress), for example, a couple of joint works (with Abel and Greig) on BIBD's and a singly authored paper on double coverings.

I am fascinated by the object of my studies and research and have many ideas for future projects and experiments. Needless to say, I am always trying to see the applications of what I am doing, or, alternatively, to look for solutions of practical problems. I believe that I have very good organizational skills, and I am able to conduct and supervise research, collaborate with other researchers, and communicate the results successfully to a broader audience. I also believe that I will be able to make other valuable contributions to the Mathematical Science in the future.