Two cyclic arrangement problems in finite projective geometry : parallelisms and two-intersection sets
Clinton Thomas White · CaltechTHESIS (California Institute of Technology) · 2006
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.