Perfect Baer subplane partitions and three-dimensional flag-transitive planes

The classification of perfect Baer subplane partitions of PG(2, q(2)) is eq uivalent to the classification of 3-dimensional flag-transitive planes whos e translation complements contain a linear cyclic group acting regularly on the line at infinity. Since all known flag-transitive planes admit a trans lation complement containing a linear cyclic subgroup which either acts reg ularly on the points of the line at infinity or has two orbits of equal siz e on these points, such a classification would be a significant step toward s the classification of all 3-dimensional flag transitive planes. Using lin earized polynomials, a parametric enumeration of all perfect Baer subplane partitions for odd q is described. Moreover, a cyclotomic conjecture is giv en, verified by computer for odd prime powers q < 200, whose truth would im ply that all perfect Baer subplane partitions arise from a construction of Kantor and hence the corresponding flag-transitive planes are all known.