Significant Examples of Scholarly Work

[1] A.H.Baartmans, I.D.Bluskov, V.D.Tonchev, The Preparata Codes and a Class of 4-Designs, Journal of Combinatorial Designs, vol.2, No.3(1994), 167-170.

[2] I. Bluskov, Some t-Designs are Minimal (t+1)-Coverings, Discrete Mathematics 188(1998), 245-251.

[3] I. Bluskov and K. Heinrich, General Upper Bounds on The Minimum Size of Covering Designs, Journal of Combinatorial Theory, Series A, 86(1999), 205-213.

[4] I. Bluskov, M. Greig and K. Heinrich, Infinite Classes of Covering Numbers, Canadian Mathematical Bulletin, 43(2000), 385-396.

[5] J. Abel, I. Bluskov, M. Greig, Balanced Incomplete Block Designs with Block Size 8, Journal of Combinatorial Designs 9(2001), 233-268.

[6] I. Bluskov, Level-Dependent Experimental Optimization, Journal of Combinatorial Mathematics and Combinatorial Computing, 42(2002), 139-145.

[7] J. Abel, I. Bluskov, M. Greig, Balanced Incomplete Block Designs with Block Size 9: Part II, Discrete Mathematics, 279(2004), 5-32.

[8] J. Abel, A. Assaf, F. Bennet, I. Bluskov and M. Greig, Pair Covering Designs with Block Size 5, accepted for publication in Discrete Mathematics, 307(2006), 1776-1791.

Significance

The results in these papers have been presented in conference talks and have been a subject of considerable interest to my colleagues.

In [1], we proved an extension theorem for t-designs. We then applied the new theorem to construct a family of 4-(4^m+1,5,2) designs by extending designs related to the 3-designs formed by the minimum weight vectors in the Preparata code of length n = 4^m, m ≥ 2. At that time, the only known construction of a 3-(16,6,4) design with minimum block distance 6 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 studied it with Baartmans. They managed to see a connection with the Preparata codes and expand my construction to the general family This was the first example of an infinite family of designs with t ≥ 4 and constant λ. Recently, Bierbrauer found other examples, but our family is still the only example of an infinite family of designs with parameter t ≥ 4 obtained from codes. It is also the only family of designs with the smallest λ 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], I proved the existence of three new families of minimal (t + 1) - coverings obtained from t - designs. The coverings produce new covering numbers for infinite classes of parameters. Aside from t - designs (which are also t - coverings), prior to this work only three infinite families of coverings with t ≥ 3 were known, those obtained by Ray-Chaudhuri in 1968, Abraham, Ghosh and Ray-Chaudhuri in 1968, and Todorov in 1984. The proofs of the existence of two of the families were obtained as a result of a study of the intersection numbers of families of designs, an approach that has been employed in the construction of coverings for the first time.

In [3], we extended some results from our previous work and works of other researchers, and found better upper bounds on the covering numbers for several infinite families of parameters. Two of the results were based on the remarkable proof of Preparata to the fact that the binary Hamming code decomposes into translates of the Preparata code. One of the families produced coverings which were asymptotically minimal.

Paper [4] contains proofs of the existence of several new infinite classes of covering numbers. Some of the classes depend on two or three different parameters, so that any particular value of one of the parameters produces a separate infinite class, thereby resolving the existence of minimum coverings for quite a wide spectrum of parameters. The paper introduced a new technique in the construction of coverings. The results of this paper have been frequently cited in recent papers on coverings.

In [5], we resolved, with a few exceptions, the question of existence of balanced incomplete block designs with block size 8. This area of research has been virtually stagnant since the works of Hanani on the cases of block size up to 6 in the 1970's. Our research started while I was experimenting with combinatorial optimization. In the process, we resolved several infinite series of parameters within the main case of block size 8. These were the first completely resolved infinite series since those found by Hanani for smaller block size cases. The methods and techniques developed in this paper gave us a good start for extensive work on the next open case of BIBD (block size 9), which resulted in three papers by the same research team (Abel, myself and Greig).

These three papers completely resolved (with a small number of exceptions) the question of existence of balanced incomplete block designs with block size 9.

Paper [6] came as a result of my interests 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 my results in recent papers ([5], for example) were obtained by optimization. This work gave me the opportunity to study the existing optimization algorithms, compare their performance, notice some common weaknesses, and eventually develop a new algorithm that overperforms (and, in fact, generalizes) all of the existing ones. I call it Level Dependent Experimental Optimization (LDEO). The paper [6] contains a description of the new algorithm. The significance of this algorithm is that, unlike any other, LDEO is a self-improving algorithm, where the self-improvability is based only on the particular combinatorial optimization problem being solved (as opposed to being based on a prespecified function used in other methods, such as simulated annealing, for example).

Using the momentum developed during our successful work on BIBDs of block size 8 [5], we proceeded with attacking the open cases of BIBDs with block size 9. The paper [7] is the second of a sequence of three papers written by the same authors (Abel, Greig and myself), and dealing with BIBDs of block size 9. In [7] we studied BIBDs with k = 9 and λ = 3, 6 and 12. We showed that the necessary conditions on v are sufficient, with the possible exceptions of v = 177, 345, 385 when λ = 3, and v = 213 when λ = 6. We also gave constructions of TD₃ (10,n)s with 13 possible exceptions, and reduced the number of unknown TDs with block sizes 8 and 9.

The major work on coverings of block size 5 [8] was initiated in 2003 by A. Assaf, M. Greig, and myself. Subsequently, F. Bennett and J. Abel joined the group. For fixed v, the covering number C(v,5,2) is the number of blocks in any minimum 2-(v,5,1) covering. For v ≤ 24, the covering numbers C(v,5,2) are known, and all but C(8,5,2) exceed the Schonheim bound. However, for all v ≥ 28, it seemed probable that C(v,5,2) meets the Schonheim bound. We studied pair covering designs with a block size of 5 and v\equiv 0 \pmod 4. The other classes modulo 4 were resolved in a series of papers by a number of researchers. The cases v ≡ 2,3 mod 4 were completely resolved in or before 1998; the last cases were done by Lamken, Mills, Mullin and Vanstone. The case v ≡ 1 mod 4 was resolved at the same time with few exceptions left. The list of exceptions was recently reduced in a series of papers by Abel, Assaf, Bennett, Chang, Ge, Greig and Ling. The case v ≡ 0 mod 4 was, by far, the most difficult. There were only few sporadic values of v for which the covering number was known. We were able to completely resolve this case for all but 17 possible exceptional values lying in the range 40 ≤ v ≤ 280. We also updated the existence status of (v,5, λ) RBIBDs, giving new (225,5,1) and (90,5,4) RBIBDs.