Citation
AjoodaniNamini, Shahin (1998) Large Sets of tDesigns. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/y84mhs59. https://resolver.caltech.edu/CaltechTHESIS:11202019172054366
Abstract
We investigate the existence of large sets of tdesigns. We introduce twise equivalence and (n, t)partitionable sets. We propose a general approach to construct large sets of tdesigns. Then, we consider large sets of a prescribed size n. We partition the set of all ksubsets of a vset into several parts, each can be written as product of two trivial designs. Utilizing these partitions we develop some recursive methods to construct large sets of tdesigns. Then, we direct our attention to the large sets of prime size. We prove two extension theorems for these large sets. These theorems are the only known recursive constructions for large sets which do not put any additional restriction on the parameters, and work for all t and k. One of them, has even a further advantage; it increase the strength of the large set by one, and it can be used recursively which makes it one of a kind. Then applying this theorem recursively, we construct large sets of tdesigns for all t and some blocksizes k.
Hartman conjectured that the necessary conditions for the existence of a large set of size two are also sufficient. We suggest a recursive approach to the Hartman conjecture, which reduces this conjecture to the case that the blocksize is a power of two, and the order is very small. Utilizing this approach, we prove the Hartman conjecture for t = 2. For t = 3, we prove that this conjecture is true for infinitely many k, and for the rest of them there are at most k/2 exceptions.
In Chapter 4 we consider the case k = t + 1. We modify the recursive methods developed by Teirlinck, and then we construct some new infinite families of large sets of tdesigns (for all t), some of them are the smallest known large sets. We also prove that if k = t + 1, then the Hartman conjecture is asymptotically correct.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Mathematics  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  14 July 1997  
Other Numbering System: 
 
Record Number:  CaltechTHESIS:11202019172054366  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:11202019172054366  
DOI:  10.7907/y84mhs59  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  13585  
Collection:  CaltechTHESIS  
Deposited By:  Mel Ray  
Deposited On:  21 Nov 2019 01:35  
Last Modified:  19 Apr 2021 22:37 
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