Citation
White, Clinton Thomas (2002) Two cyclic arrangement problems in finite projective geometry : parallelisms and twointersection sets. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd06052006143933
Abstract
Two arrangement problems in projective geometries over finite fields are studied, each by imposing the condition that solutions be generated by some cyclic automorphism group. Part I investigates cyclic parallelisms of the lines of PG(2n  1,q). Properties of a collineation which can act transitively on the spreads of a parallelism are determined, and these are used to show nonexistence of cyclic parallelisms in the cases of PG(2n  1,q) with gcd(2n  1,q  1) > 1 and PG(3, q) with q = 0 (mod 3). Along with the result first established by Pentilla and Williams that PG (3, q) admits cyclic (and regular) parallelisms if q = 2 (mod 3), this completes the existence problem in dimension 3. Cyclic regular parallelisms of PG(3, q) are considered from the point of view of linear transversal mappings, leading to a conjectured classification. Finally, some partial results and open problems relating to cyclic parallelisms in odd dimensions greater than 3 are discussed. Part II is joint work with B. Schmidt, investigating which subgroups of Singer cycles of PG(n  1,q) have orbits which are twointersection sets. This problem is essentially equivalent to investigating which irreducible cyclic codes have at most two nonzero weights. The main results are necessary and sufficient conditions on the parameters for a Singer subgroup orbit to be a twointersection set. These conditions allow a computer search which revealed two previously known families and eleven sporadic examples, four of which are believed to be new. It is conjectured that there are no further examples.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Restricted to Caltech community only 
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Thesis Committee: 

Defense Date:  5 September 2001 
Record Number:  CaltechETD:etd06052006143933 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd06052006143933 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  2463 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  05 Jun 2006 
Last Modified:  28 Jul 2014 22:38 
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