| Abstract | U ovoj disertaciji proučavamo kvazisimetrične dizajne s iznimnim parametrima te metode za njihovu konstrukciju s pretpostavljenom grupom automorfizama. Tablica dopustivih iznimnih parametara kvazisimetričnih \(2-(v, k, \lambda)\) dizajna s presječnim brojevima x i y je proširena do v = 150 te je time dobiveno ukupno 260 skupova parametara, pri čemu za njih 172 egzistencija kvazisimetričnih dizajna nije poznata. Pomoću Kramer-Mesnerove metode – poznate metode za konstrukciju t-dizajna sa zadanom grupom automorfizama, konstruirani su mnogi dizajni s parametrima \(t-(v, k, \lambda)\) odgovarajućih kvazisimetričnih dizajna iz navedene tablice te je time pokazano koliko je problem konstrukcije 2-dizajna bez uvjeta kvazisimetričnosti lakši. U konstrukciji kvazisimetričnih dizajna sa zadanom grupom automorfizama osnovni koraci su generiranje orbita i konstrukcija dizajna iz dobivenih orbita. U disertaciji su opisani algoritmi za generiranje kratkih orbita te svih orbita podskupova zadane veličine nekog skupa pod djelovanjem pretpostavljene grupe automorfizama. Također, opisani su algoritmi na kojima se temelje metode za konstrukciju kvazisimetričnih dizajna, a to su Kramer-Mesnerova metoda, metoda temeljena na traženju klika i metoda temeljena na orbitnim matricama. Korištenjem navedenih algoritama konstruirani su novi kvazisimetrični dizajni s parametrima 2-(28, 12, 11), x = 4, y = 6, zatim 2-(36, 16, 12), x = 6, y = 8 te 2-(56, 16, 6), x = 4, y = 6. Isto tako, utvrđena je egzistencija kvazisimetričnog 2-(56, 16, 18) dizajna s presječnim brojevima x = 4 i y = 8, koja ranije nije bila poznata, te su konstruirana četiri neizomorfna dizajna s tim parametrima. Nadalje, utvrđeno je da samo već poznati kvazisimetrični dizajni s projektivnim parametrima postoje s određenim grupama automorfizama. |
| Abstract (english) | In this thesis, we study quasi-symmetric designs with exceptional parameters and methods for their construction with prescribed automorphism groups. A \(t-(v, k, \lambda)\) design is quasi-symmetric if any two blocks intersect either in x or in y points, for non-negative integers x < y. The classification of quasi-symmetric 2-designs is a difficult open problem and there are many triples \((v, k, \lambda)\) for which existence is unknown. A. Neumaier in [74] defines four classes of quasi-symmetric 2-designs: multiples of symmetric designs, strongly resolvable designs, Steiner designs, and residuals of biplanes. If a quasi-symmetric design or its complement does not belong to any of these classes, he calls it exceptional. In the same paper, he published the first table of admissible exceptional parameters of quasi-symmetric 2-designs with v < 40. Updated and extended tables were published by V. D. Tonchev, A. R. Calderbank, and M. S. Shrikhande. The last table was published in [85] and contains all exceptional parameters with v ≤ 70. The thesis is divided into three chapters. In the first chapter, we give some basic definitions, notations, and results about t-designs, quasi-symmetric t-designs, and quasisymmetric 2-designs in particular. We also explain their important connections with strongly regular graphs and self-orthogonal codes. Furthermore, we describe the known families of quasi-symmetric 2-designs. Finally, at the end of this chapter we update the table of admissible exceptional parameters with new results and extend it to v = 150. The new table contains 260 parameter sets, and for 172 parameter sets the existence of quasi-symmetric designs is unknown. In the second chapter, we explore the construction of 2-designs with prescribed automorphism groups, not necessarily quasi-symmetric. We give some basic concepts and results from group theory, in particular permutation groups, and consider (full) automorphism groups of designs. Furthermore, we describe one of the most widespread methods for the construction of designs with prescribed automorphism groups, the Kramer-Mesner method. Using this method, we construct a lot of designs with exceptional parameters \(2-(v, k, \lambda)\) of quasi-symmetric designs. The conclusion is that the construction of 2-designs is much easier without the condition of quasi-symmetry. In the last and most important chapter, we develop computational methods for the construction of quasi-symmetric designs with a prescribed automorphism group G. The v construction is done in two basic steps: firstly, generate the good orbits of G on k-element subsets of a v-element set, and secondly, select orbits comprising blocks of the design. In the first section of this chapter, we give some ideas for selecting automorphism groups. In the second section, we explain algorithms for generating orbits. Depending on the parameters of the design and size of the automorphism group, we use the algorithm for short orbits, the algorithm for all orbits, or generate orbits from orbit matrices. In the third section, we explain methods for the construction of quasi-symmetric designs from the generated orbits. We use the Kramer-Mesner method, a method based on clique search, and a method based on orbit matrices. All these methods had been previously known in design theory. We adapt them for the construction of quasi-symmetric designs and perform the constructions for some feasible parameters and groups. We increase the number of known designs with parameters 2-(28, 12, 11), x = 4, y = 6, 2-(36, 16, 12), x = 6, y = 8, 2-(56, 16, 6), x = 4, y = 6, and establish the existence of quasi-symmetric 2-(56, 16, 18) designs with intersection numbers x = 4 and y = 8. Furthermore, using binary codes associated with newly constructed 2-(56, 16, 6) designs, we find even more quasi-symmetric designs with these parameters. All the new quasi-symmetric 2-(56, 16, 18) designs can be extended to symmetric 2-(78, 22, 6) designs and in this way the number of known symmetric designs with these parameters is significantly increased. In the last section of this chapter, we attempt construction of quasi-symmetric designs with projective parameters and certain automorphism groups, but find only known examples of such designs. |
