Full Record

Author | Balachandran, Niranjan |

Title | The 3-Design Problem |

URL | http://rave.ohiolink.edu/etdc/view?acc_num=osu1211922186 |

Publication Date | 2008 |

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 |

Sample Search Hits | Sample Images

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Combinatorial Constructions . . . . . . . . . . . . . . . . . . . . . . . .
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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…