On tetrahedrally closed line systems and a generalization of the Haemers-Roos inequality

We generalize the well-known Haemers-Roos inequality for generalized hexagons of order (s, t) to arbitrary near hexagons S with an order. The proof is based on the fact that a certain matrix associated with S is idempotent. The fact that this matrix is idempotent has some consequences for tetrahedrally closed line systems in Euclidean spaces. One of the early theorems relating these line systems with near hexagons is known to contain an essential error. We fix this gap here in the special case for geometries having an order. (C) 2020 Elsevier Inc. All rights reserved

The hyperplanes of the near hexagon related to the extended ternary Golay code

We prove that the near hexagon associated with the extended ternary Golay code has, up to isomorphism, 25 hyperplanes, and give an explicit construction for each of them. As a main tool in the proof, we show that the classification of these hyperplanes is equivalent to the determination of the orbits on vectors of certain modules for the group 2 center dot M-12

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page