On the intersection of distance--ovoids and subpolygons of generalized polygons

In [A. De Wispelaere, H. Van Maldeghem, Regular partitions of (weak) finite generalized polygons, Des. Codes Cryptogr. 47 (2008) 53–73] (see also [A. De Wispelaere, Ovoids and spreads of finite classical generalized hexagons and applications, Ph.D. Thesis, Ghent University, 2005]), a technique was given for calculating the intersection sizes of combinatorial substructures associated with regular partitions of distance-regular graphs. This technique was based on the orthogonality of the eigenvectors which correspond to distinct eigenvalues of the (symmetric) adjacency matrix. In the present paper, we give a more general method for calculating intersection sizes of combinatorial structures. The proof of this method is based on the solution of a linear system of equations which is obtained by means of double countings. We also give a new class of regular partitions of generalized hexagons and determine under which conditions ovoids and subhexagons of order ( s ′ , t ′ ) of a generalized hexagon of order s intersect in a constant number of points. If the automorphism group of the generalized hexagon is sufficiently large, then this is the case if and only if s = s ′ t ′ . We derive a similar result for the intersection of distance-2-ovoids and suboctagons of generalized octagons.