Pairwise balanced designs and related codes

Abstract

This thesis deals with pairwise balanced designs, group divisible designs and related codes. We study pairwise balanced designs with three consecutive block sizes. In particular, we investigate the spectrum of pairwise balanced designs with block sizes give, six and seven; six, seven and eight; seven, eight and nine; and eight, nine and ten. We have standardized the known techniques for constructing pairwise balanced designs with consecutive block sizes. New constructions employing certain line configurations in finite projective planes are also developed. The direct and recursive constructions both require the existence of finite projective planes, particularly goals in desarguesian projective planes. Combining known and new techniques, we have essentially determined the spectra for these pairwise balanced designs.

We also study uniform group divisible designs with block size five. We prove that uniform group divisible designs with block size five exists for all parameters satisfying the basic necessary conditions with a finite number of possible exceptions. Many of direct constructions are required to obtain this strong existence result. In particular, we have constructed many group divisible designs with block five admitting a large automorphism group. Several new recursive constructions are presented and used to settle this problem. One recursive construction requires a relatively new type of combinatorial design, the modified group divisible design, which is also studied in this thesis.

Finally, we study some coding theoretic problems arising from computer science which have design theoretic connections. We have related a well known problem in combinatorial design theory and finite geometry to coding theory. The existence of anti-Pasch Steiner triple systems corresponds to the existence of a certain type of erasure code. We generalize the existing constructions to obtain some new infinite classes of anti-Pasch Steiner triple systems. In addition, we study some related problems concerning Steiner triple systems avoiding certain configurations.