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Author
Title The 3-Design Problem
URL
Publication Date
Degree PhD
Discipline/Department Mathematics
Degree Level doctoral
University/Publisher The Ohio State University
Abstract This dissertation studies the ‘asymptotic existence’ conjecture for 3-designs with the primary goal of constructing new families of 3-designs. More specifically, this dissertation includes the following: Firstly, by considering the action of the group PSL(2,q) on the finite projective line and the orbits of the action of this group to construct simple 3-designs. While the case q congruent to 3 modulo 4 is 3-homogeneous (so that orbits of any ‘base’ block’ would yield designs), the case q congruent to 1 modulo 4 does not work the same way. We overcome some of these issues by considering appropriate unions of orbits to produce new infinite families of 3-designs with PSL(2,q) acting as a group of automorphisms. We also prove that our constructions actually produce an abundance of simple 3-designs for any block size if q is sufficiently large and also construct a large set of Divisible designs as an application of our constructions. We generalize the notion of a Candelabra system to more general structures, called Rooted Forest Set systems and prove a few general results on combinatorial constructions for these general set structures. Then, we specialize to the case of k=6 and extend a theorem of Hanani to produce new infinite families of Steiner 3-designs with block size 6. Finally, we consider Candelabra systems and prove that a related incidence matrix has full row rank over the rationals. This leads to interesting possibilities for ‘lambda large’ theorems for Candelabra systems. While a ‘lambda large’ theorem for Candelabra systems do not directly yield any Steiner 3-design, it allows for constructions of new Steiner 3-designs on large sets using methods such as Block spreading.
Subjects/Keywords Mathematics; 3-design; Candelabra system; projective special linear groups
Contributors Robertson, Neil (Advisor)
Language en
Rights unrestricted ; This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws.
Country of Publication us
Format application/pdf
Record ID oai:etd.ohiolink.edu:osu1211922186
Repository ohiolink
Date Indexed 2020-10-19
Grantor The Ohio State University

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…28 38 43 46 62 65 Combinatorial Constructions . . . . . . . . . . . . . . . . . . . . . . . . 67 3.1 3.2 Classical 3-transitive groups The Groups PSL(2, q), q odd The case q ≡ 3 (mod 4) . . . The case q ≡ 1 (mod 4)…

…v, k, λ) design exists. The problem of construction asks for (if possible efficient) a constructive solution to the existence problem. The problem of Symmetries involves constructions for designs with large automorphism groups and the…

…these constructions bear a distinct algebraic flavor in the sense that the underlying set upon which the design is constructed has a nice algebraic structure (Groups, fields, rings, etc). One may then use other (combinatorial) methods…

…family of groups to use this technique effectively. An attempt to describe all possible algebraic tools that may be employed to construct 3-designs is an exercise in futility, so we restrict our attention to transitive (multiplytransitive)…

…actions of groups in this thesis. One such interesting family of groups is PGL(2, q), for q a prime power. Returning to an earlier point, the fact that there are infinitely many 3-designs (in fact, Steiner designs) with block size 5 is…

…1 = q(q 2 − 1). Remark: While there exist several groups that are 3-transitive (for instance the permutation groups Sn ), it is also necessary that the groups are not ‘too large’ since that would imply that λ grows large as well…

…We will see this in more detail in the next chapter. Another closely related family of sharply 3-transitive groups are the ‘twisted’ PGL(2, q 2 ) with q being a prime power. This group consists of two classes of mappings on the finite…

…as the theorem stated above. However, in terms of parameters, these designs are still 3 − (q 2 + 1, q + 1, 1). The following theorem which was proved by Zassenhaus in 1938 describes the set of all sharply 3-transitive groups and the…

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