# Generalization and refinement of some algorithms for construction and substructures investigation of block designs

Dumičić Danilović, Doris (2014) Generalization and refinement of some algorithms for construction and substructures investigation of block designs. Doctoral thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

In this thesis we describe the algorithms for construction of block designs admitting an action of an automorphism group using orbit matrices. We develop and describe a program which constructs mutually nonisomorphic orbit matrices for arbitrary block design and automorphism group, which is a generalisation of the program for obtaining orbit matrices for some symmetric design and automorphism group described in the paper [21]. The second step in the construction is often called an indexing of orbit matrices, respectively construction of block designs from orbit matrices. Indexing often lasts too long, therefore we develop an algorithm for the refinement of orbit matrices, based on the application of the composition series of the solvable automorphism group which acts on design. Mentioned algorithms are used as a base for computer programs written in program GAP. Classification of all 2-(45, 5, 1) designs admitting an action of an automorphism of order 6 and classification of all 2-(45, 12, 3) designs admitting an action of an involution is presented. Moreover, we have constructed all, up to isomorphism, 2-(45, 5, 1) designs admitting an action of group $Z_{3}\times Z_{3}$ in the cases when there exist two subgroups of order 3 in the group $Z_{3}\times Z_{3}$ that act on the design having no common point and block orbits of length 3. Besides this, we have constructed 2-(45, 5, 1) designs admitting an action of group $S_{3}$. The classification of all 2-(78, 22, 6) admitting an action of group $G \cong F rob_{39} \times Z_{2}$ is presented. In this case, the algorithm for the refinement of orbit matrices is based on the application of the principal series of the group G. An important result of the thesis is the proof that there does not exists (78,22,6) difference set in the group G. Besides that, the subject of the thesis is to examine the existence of certain subdesignes in a block designs using an incomplete search with a modified genetic algorithm, in case when complete methods (exhaustive search) do not give enough or any results in a reasonable period of time. Therefore modified genetic algorithms for finding subdesigns probing incidence matrices for some block designs we develop and describe. Modified genetic algorithm for finding unitals in symmetric designs is applied on search for unitals in symmetric designs with parameters 2-(66,26,10) and 2-(36,15,6). Moreover, modified genetic algorithm for finding subdesigns with given parameters in arbitrary block design is applied on search for 2-(11, 5, 2) subdesigns in 2-(66, 26, 10) designs.