Group Divisible 3-Wise Balanced Designs: Theory And Applications

Group divisible t-wise balanced designs are of utmost importance in combinatorial design theory,and have been widely used in many areas.For t=2, group divisible designs were an essential ingredient in the recursive constructions used in the seminal works of Wilson and Hanani(two of the founders of combinatorial design theory),which established necessary and sufficient conditions for the existence of pairwise balanced designs.Much work has been done on such designs.For t=3,two definitions for 3-analogues of group divisible designs -candelabra quadruple systems and group divisible 3-designs were first. introduced by Hanani in 1963.In 1994.Hartman gave a more comprehensive account of the 3-analogues of group divisible designs,which were applicable for the generalization of Wilson's(and Hanani's) fundamental constructions to produce 3-wise balanced designs.In these 3-analogues of the fundamental constructions, group divisible 3-designs(called H-designs in the sequel) are also used as essential ingredients.The research on Steiner quadruple systems-a special class of H-designs with each group of size one can be traced back to 1840s.The first and second complete proofs for the existence of such designs were given by Hanani in 1960s. All the existing proofs are rather cumbersome,even though simplified proofs have been given by Lenz in 1985 and by Hartman in 1994.For the existence of Steiner quadruple systems with resolvability,the known complete solution was obtained by a joint effort of Hartman.Ji and Zhu over twenty years long.As for the general existence of resolvable H-designs,however,not much is known.In Chapters 2 and 3 of this dissertation,not only do we provide alternative existence proofs for Steiner quadruple systems and resolvable Steiner quadruple systems,we give an almost complete solution to the general existence problem of resolvable H-designs and construct several infinite classes of nonuniform H-designs.As a consequence of the existence result of resolvable H-designs,we establish the necessary aud sufficient,conditions for the existence of resolvable G-designs in Chapter 2,which is another kind of 3-analogue of group divisible designs.As a byproduct,we also show the existence of maximal resolvable packings of triples by quadruples, minimal resolvable coverings of triples by quadruples and a class of uniformly resolvable Steiner systems.As applications of the theory of group divisible 3-wise balanced designs,two open problems in group testing and optical networks are also discussed.First, we give a complete solution to the problem posed by Jimbo et al.on the block sequences of Steiner quadruple systems with error correcting consecutive unions in Chapter 4.Such sequences are useful when considering the error detecting and correcting capability of combinatorial group testing for consecutive positives, which is essential in view of applications such as DNA library screening. Then in Chapter 5,we investigate the design of fault-tolerant routings with levelled minimum optical indices,which plays an important role in the wavelength division multiplexing optical networks.By introducing the new concept of a.large set of even levelled(?)-design,we solve nearly one-third of the existence problem for optimal routings with levelled minimum optical indices based on the theory of 3-wise balanced designs and partitionable candelabra systems.