On the universal embedding of the near hexagon related to the extended ternary Golay code

AbstractLet E1 be the near hexagon on 729 points related to the extended ternary Golay code. We prove in an entirely geometric way that the generating and embedding ranks of E1 are equal to 24. We also study the structure of the universal embedding e˜ of E1. More precisely, we consider several nice subgeometries A of E1 and determine which kind of embedding e˜A is, where e˜A is the embedding of A induced by e˜

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

The hyperplanes of the U (4)(3) near hexagon

Combining theoretical arguments with calculations in the computer algebra package GAP, we are able to show that there are 27 isomorphism classes of hyperplanes in the near hexagon for the group U (4)(3). We give an explicit construction of a representative of each class and we list several combinatorial properties of such a representative

The hyperplanes of the glued near hexagon Q(5,2) ⊗ Q(5,2)

With the aid of the computer algebra system GAP, we show that the glued near hexagon Q(5, 2) circle times Q(5, 2) has 16 isomorphism classes of hyperplanes. We give at least one explicit construction for a representative of each isomorphism class and we list several properties of such a representative

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

The valuations of the near polygon n

We show that every valuation of the near 2n-gon G(n), n >= 2, is induced by a unique classical valuation of the dual polar space DH(2n - 1, 4)into which G(n) is isometrically embeddable

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic