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New Geometric Large Sets
 Date Issued:
 2016
 Summary:
 Let V be an ndimensional vector space over the field of q elements. By a geometric t[q^n, k, λ] design we mean a collection D of kdimensional subspaces of V, called blocks, such that every tdimensional subspace T of V appears in exactly λ blocks in D. A large set, LS [N] [t, k, q^n], of geometric designs is a collection on N disjoint t[q^n, k, λ] designs that partitions [V K], the collection of kdimensional subspaces of V. In this work we construct nonisomorphic large sets using methods based on incidence structures known as the KramerMesner matrices. These structures are induced by particular group actions on the collection of subspaces of the vector space V. Subsequently, we discuss and use computational techniques for solving certain linear problems of the form AX = B, where A is a large integral matrix and X is a {0,1} solution. These techniques involve (i) lattice basisreduction, including variants of the LLL algorithm, and (ii) linear programming. Inspiration came from the 2013 work of Braun, Kohnert, Ostergard, and Wassermann, [17], who produced the first nontrivial large set of geometric designs with t ≥ 2. Bal Khadka and Michael Epstein provided the knowhow for using the LLL and linear programming algorithms that we implemented to construct the large sets.
Title:  New Geometric Large Sets. 
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Name(s): 
Hurley, Michael Robert, author Magliveras, Spyros S., Thesis advisor Florida Atlantic University, Degree grantor Charles E. Schmidt College of Science Department of Mathematical Sciences 

Type of Resource:  text  
Genre:  Electronic Thesis Or Dissertation  
Date Created:  2016  
Date Issued:  2016  
Publisher:  Florida Atlantic University  
Place of Publication:  Boca Raton, Fla.  
Physical Form:  application/pdf  
Extent:  84 p.  
Language(s):  English  
Summary:  Let V be an ndimensional vector space over the field of q elements. By a geometric t[q^n, k, λ] design we mean a collection D of kdimensional subspaces of V, called blocks, such that every tdimensional subspace T of V appears in exactly λ blocks in D. A large set, LS [N] [t, k, q^n], of geometric designs is a collection on N disjoint t[q^n, k, λ] designs that partitions [V K], the collection of kdimensional subspaces of V. In this work we construct nonisomorphic large sets using methods based on incidence structures known as the KramerMesner matrices. These structures are induced by particular group actions on the collection of subspaces of the vector space V. Subsequently, we discuss and use computational techniques for solving certain linear problems of the form AX = B, where A is a large integral matrix and X is a {0,1} solution. These techniques involve (i) lattice basisreduction, including variants of the LLL algorithm, and (ii) linear programming. Inspiration came from the 2013 work of Braun, Kohnert, Ostergard, and Wassermann, [17], who produced the first nontrivial large set of geometric designs with t ≥ 2. Bal Khadka and Michael Epstein provided the knowhow for using the LLL and linear programming algorithms that we implemented to construct the large sets.  
Identifier:  FA00004732 (IID)  
Degree granted:  Dissertation (Ph.D.)Florida Atlantic University, 2016.  
Collection:  FAU Electronic Theses and Dissertations Collection  
Note(s):  Includes bibliography.  
Subject(s): 
Group theory. Finite groups. Factorial experiment designs. Irregularities of distribution (Number theory) Combinatorial analysis. 

Held by:  Florida Atlantic University Libraries  
Sublocation:  Digital Library  
Links:  http://purl.flvc.org/fau/fd/FA00004732  
Persistent Link to This Record:  http://purl.flvc.org/fau/fd/FA00004732  
Use and Reproduction:  Copyright © is held by the author with permission granted to Florida Atlantic University to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.  
Use and Reproduction:  http://rightsstatements.org/vocab/InC/1.0/  
Host Institution:  FAU  
Is Part of Series:  Florida Atlantic University Digital Library Collections. 