Construction and non-embeddability of quasi-residual designs

The notions of residual and derived designs were first introduced into design theory in a seminal paper by Bose (1939). Quasi-residual and quasi-derived designs are defined to be 2-designs with the parameters of residual and derived designs of a symmetric design. If a quasi-residual (resp. a quasi-derived) design is in fact a residual (resp. a derived) design of a symmetric design, then it is called embeddable. Otherwise it is said to be non-embeddable. The embedding problem of quasi-residual and quasi-derived design into a symmetric design is an old and natural question. Bhattacharya (1944) gave the first example of quasi-residual design which is non-embeddable. The goal of this dissertation is to obtain constructions of these types of designs, and also to investigate their non-embeddability.; Chapter II is devoted to known results on quasi-residual and quasi-derived designs. Chapter III deals with problems concerning quasi-derived designs. We give a method of constructing a quasi-derived design having a maximal subdesign using alpha-resolvable designs. We show that such a design is non-embeddable if the maximal subdesign satisfies a certain divisibility condition. Also, alpha-resolvable group divisible designs are used to construct quasi-derived designs with non-maximal subdesigns. We further show that such designs are non-embeddable provided that the non-maximal subdesign satisfies a certain inequality condition. Using these techniques, we obtain several new families of non-embeddable quasi-derived designs.; A challenging question is to determine the existence or non-existence of nonembeddable quasi-residual (quasi-derived) designs with parameters corresponding to a known family of symmetric designs. Chapters IV and V are devoted to partially answering this question for Hadamard designs and Menon designs. Using a computer, we apply a method due to Tonchev [55] to construct designs with small parameters that correspond to Hadamard or Menon designs. We then use recursive methods in order to obtain infinite families of non-embeddable Hadamard and Menon designs.; In Chapter VI we discuss two further construction techniques. One of these methods uses generalized Hadamard matrices, and the other makes use of transversal designs. These techniques are generalizations of results in [30] and [56]. As a consequence, we construct further families of non-embeddable quasi-residual and quasi-derived designs. We also obtain 20 non-isomorphic 2 - (36,12,11) designs. This improves the lower bound on the number of designs with such parameters mentioned in [12].; Chapter VII is a summary of all results known to us on quasi-residual and quasi-derived designs. We include two tables of possible parameter sets of quasi-residual designs with replication number r ≤ 41 We also list families of non-embeddable quasi-residual and quasi-derived designs that have been constructed in this dissertation or in the literature. In the last section of this chapter, we make some remarks on quasi-residual and quasi-derived designs with parameters corresponding to known families of symmetric designs.