White, Clinton T. (2002) Two cyclic arrangement problems in finite projective geometry : parallelisms and two-intersection sets. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-06052006-143933
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 two-intersection sets. This problem is essentially equivalent to investigating which irreducible cyclic codes have at most two non-zero weights. The main results are necessary and sufficient conditions on the parameters for a Singer subgroup orbit to be a two-intersection 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|
|Thesis Availability:||Restricted to Caltech community only|
|Defense Date:||5 September 2001|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Imported from ETD-db|
|Deposited On:||05 Jun 2006|
|Last Modified:||26 Dec 2012 02:51|
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