The Distance-2-Sets of the Slim Dense Near Hexagons

Bart De Bruyn and Pieter Vandecasteele

Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, B-9000 Gent, Belgium

{bdb,pvdecast}@cage.ugent.be

Annals of Combinatorics 10 (2) p. 193-210 June, 2006

Abstract:

We determine all distance-2-sets and all valuations of the 11 slim dense near hexagons. We consider this as a necessary step in order to classify all slim dense near octagons. These near octagons will be classified in a forthcoming paper.

References:

1. M. Aschbacher, Flag structures on Tits geometries, Geom. Dedicata 14 (1983) 21–32.

2. S. Ball, P. Govaerts, and L. Storme, On ovoids of parabolic quadrics, Des. Codes Cryptogr. 38 (1) (2006) 131–145.

3. A.E. Brouwer and H.A. Wilbrink, The structure of near polygons with quads, Geom. Dedicata 14 (1983) 145–176.

4. A.E. Brouwer, A.M. Cohen, J.I. Hall, and H.A.Wilbrink, Near polygons and Fischer spaces, Geom. Dedicata 49 (1994) 349–368.

5. P.J. Cameron, Dual polar spaces, Geom. Dedicata 12 (1982) 75–86.

6. H.S.M. Coxeter, Twelve points in PG(5, 3) with 95040 self-transformations, Proc. R. Soc. A 247 (1958) 279–293.

7. B. De Bruyn, On near hexagons and spreads of generalized quadrangles, J. Algebraic Combin. 11 (2000) 211–226.

8. B. De Bruyn, On the number of nonisomorphic glued near hexagons, Bull. Belg. Math. Soc. Simon Stevin 7 (2000) 493–510.

9. B. De Bruyn, On the finiteness of near polygons with 3 points on every line, J. Algebraic Combin. 18 (2003) 41–46.

10. B. De Bruyn and F. De Clerck, On linear representations of near hexagons, European J. Combin. 20 (1999) 45–60.

11. B. De Bruyn and P. Vandecasteele, Valuations of near polygons, Glasg. Math. J. 47 (2005) 347–361.

12. B. De Bruyn and P. Vandecasteele, The valuations of the near hexagons related to the Witt designs S(5, 6, 12) and S(5, 8, 24), J. Combin. Des. 14 (2006) 214–228.

13. B. De Bruyn and P. Vandecasteele, The classification of the slim dense near octagons, European J. Combin., to appear.

14. D. Pasechnik, Extending polar spaces of rank at least 3, J. Combin. Theory Ser. A 72 (1995) 232–242.

15. S.E. Payne and J.A. Thas, Finite Generalized Quadrangles, Research Notes in Mathematics, Vol. 110, Pitman, Boston, 1984.

16. E.E. Shult and A. Yanushka, Near n-gons and line systems, Geom. Dedicata 9 (1980) 1–72.