engleski

Kramer-Mesner with tactical decomposition

A t-(v, k, l) design is a finite incidence structure consisting of v points and a number of blocks (sets of points), such that each block contains exactly k points and every set of t distinct points is contained in exactly blocks. Although there are many known examples of t -designs, finite projective planes being one of them, for many parameters the question of existence remains open. In order to construct new t-(v, k, l) designs, it is practically impossible to complete an exhaustive search because the problem is of exponential complexity. It is necessary to add constraints to the search. We shall present a new approach in construction of t-(v, k, l) designs. In the last few years, my supervisor Mario-Osvin Pavcevic and Vedran Krcadinac successfully combined the well known Kramer- Mesner method and tactical decomposition and indexing approach in order to construct new t- (v, k, l) designs admitting an action of an automorphism group. In the past, tactical decomposition and indexing have been used for sporadic constructions of 2-designs. On the other hand, the Kramer-Mesner algorithm was broadly used for the construction of t-designs. It is now clear that information provided by tactical decomposition matrices can enhance the Kramer-Mesner method. This new combination of two approaches can in many cases dramatically reduce the size of the Kramer-Mesner matrix and therefore t-designs can be constructed faster and more easily. Moreover, this new method can also be used to construct other combinatorial structures with weaker properties, like symmetric configurations. We shall present an outline of this new technique as well as some new results for t-designs.

design ; tactical decomposition

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