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Construction and resolution of quadruple systems

A necessary condition for the existence of a QS(7) is that V = 2 or It (mod 6) or, trivially, V = 0 or 1 . In I960 Hanani proved that these conditions are also sufficient. He used six recursive constructions and direct constructions of a QS(l't) and a QS(38) to prove this result. The first part of this thesis contains some new recursive constructions for quadruple systems, and a unified treatment of all the existing constructions. These results enable us to give a new proof of Hanani's result using only two recursive constructions, at the expense of increasing the number of initial designs necessary for the induction. New results on the problem of embedding quadruple systems in larger quadruple systems are also obtained in this part of the thesis.