Two cyclic arrangement problems in finite projective geometry : parallelisms and two-intersection sets

Citation

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

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 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.

Thesis Files