Optimality and Constructions of cyclically resolbable BIB designs and split-block designs

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In this research project, we treated two main subjects related to design of experiment and combinatorial designs. First subject is concerned with the optimality and constructions of split-block designs. We investigated the combinatorial property of the universally optimal split-block designs and gave a construction of such designs. Such optimal design is named a balanced incomplete split-block design. Furthermore, we defined a semi-balanced incomplete split-block designs which is nealy optimal and which have lesser number of replications than the optimal designs. And we obtained constructions of semi-balanced incomplete split-block designs.The second subject is concerned with the cyclic resolvability of the BIB design generated by the finite projective geometry. We showed that the BIB design generated by points and lines of PG (7,2) has a cyclic resolution. Moreover, it was shown that the BIB design generated by points and planes of PG (5,2) has a cyclic resolution, which is the first result that there is a cyclic resolution in the BIB designs generated by points and planes of the finite projective geometry.