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The 3-design problem.

The Ohio State University.

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The 3-design problem.

Record Type:	Electronic resources : Monograph/item
Title/Author:	The 3-design problem./
Author:	Balachandran, Niranjan.
Description:	126 p.
Notes:	Adviser: Neil Robertson.
Contained By:	Dissertation Abstracts International69-05B.
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3313003
ISBN:	9780549638780

The 3-design problem.
Balachandran, Niranjan.

The 3-design problem. - 126 p.

Adviser: Neil Robertson.

Thesis (Ph.D.)--The Ohio State University, 2008.

The theory of Combinatorial designs is one of the oldest and richest disciplines in Combinatorics and has wide ranging applications in as diverse fields as Cryptography, Optics, Discrete Tomography, data structures and computer algorithms, hardware design, Interconnection networks, VLSI testing, Astronomical Imaging, and Neutron Spectroscopy and also contributes to other disciplines of Mathematics such as The theory of Unimodular lattices, Coding Theory, Computational Group theory, and Discrete and Combinatorial Geometry.

ISBN: 9780549638780Subjects--Topical Terms:

515831
Mathematics.

The 3-design problem.
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020 $a 9780549638780
035 $a (UMI)AAI3313003
035 $a AAI3313003
040 $a UMI $c UMI
100 1 $a Balachandran, Niranjan. $3 1035804
245 1 4 $a The 3-design problem.
300 $a 126 p.
500 $a Adviser: Neil Robertson.
500 $a Source: Dissertation Abstracts International, Volume: 69-05, Section: B, page: 3027.
502 $a Thesis (Ph.D.)--The Ohio State University, 2008.
520 $a The theory of Combinatorial designs is one of the oldest and richest disciplines in Combinatorics and has wide ranging applications in as diverse fields as Cryptography, Optics, Discrete Tomography, data structures and computer algorithms, hardware design, Interconnection networks, VLSI testing, Astronomical Imaging, and Neutron Spectroscopy and also contributes to other disciplines of Mathematics such as The theory of Unimodular lattices, Coding Theory, Computational Group theory, and Discrete and Combinatorial Geometry.
520 $a A t--(v, k, lambda) design is a pair (X, B ), where X is a set of size v and B is a collection of subsets of X of size k each such that every t-subset of X is contained in precisely lambda members of B . A t--(v, k, lambda) design is also denoted Slambda(t, k, v). If lambda = 1 then it is called a t-Steiner design and is denoted by S(t, k, v).
520 $a The problem of characterizing all triples (v, k, lambda) for which a t--(v, k, lambda) design exists is one of the fundamental problems in the theory of Combinatorial designs. Clearly, these parameters cannot be chosen independent of one another since there are certain necessary conditions that are to be met in order that a design exists. These are known as the arithmetic conditions or the 'admissibility conditions'.
520 $a While the admissibility conditions are necessary, they are also not sufficient; there exist several cases of parameters that satisfy the admissibility conditions and yet no design with these parameters exists. However, if the point set is large, then it is conjectured that the admissibility conditions would be sufficient as well. This is known as the 'v-large existence conjecture' or the 'asymptotic existence' conjecture. The 'asymptotic existence' conjecture has been proved for t = 2 by Wilson, following the work of several including R.C. Bose, Marshall Hall, Jr., Haim Hanani, and Dijen Ray-Chaudhuri.
520 $a This dissertation studies the 'asymptotic existence' conjecture in the specific case t = 3 with the primary goal of constructing new families of 3-designs. More specifically, this dissertation includes the following: (1) 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 &equiv; 3 (mod 4) is 3-homogeneous (so that orbits of any 'base' block' would yield designs), the case q &equiv; 1 (mod 4) does not work the same way. We however 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. We also construct a large set of Divisible designs as an application of our constructions. (2) 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 several new infinite families of Steiner 3-designs with block size 6. (3) Finally, we consider Candelabra systems and prove that a related incidence matrix has full row rank over Q . 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 (in fact, even simple 3 designs), it allows for constructions of new Steiner 3-designs on large sets following the methods of Block spreading.
590 $a School code: 0168.
650 4 $a Mathematics. $3 515831
690 $a 0405
710 2 $a The Ohio State University. $3 718944
773 0 $t Dissertation Abstracts International $g 69-05B.
790 $a 0168
790 1 0 $a Robertson, Neil, $e advisor
791 $a Ph.D.
792 $a 2008
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3313003