Abstract
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 \(2 \cdot M_{12}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brouwer, A.E.: The uniqueness of the near hexagon on 729 points. Combinatorica 2, 333–340 (1982)
Brouwer, A.E., Cohen, A.M., Hall, J.I., Wilbrink, H.A.: Near polygons and Fischer spaces. Geom. Dedic. 49, 349–368 (1994)
Brouwer, A.E., Cuypers, H., Lambeck, E.W.: The hyperplanes of the \(M_{24}\) near polygon. Gr. Comb. 18, 415–420 (2002)
Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press, Oxford (1985)
Coxeter, H.S.M.: Twelve points in \({{\rm PG}}(5,3)\) with 95040 self-transformations. Proc. R. Soc. Lond. Ser. A 247, 279–293 (1958)
De Bruyn, B.: On near-polygons and the Coxeter cap in \({{\rm PG}}(5,3)\). J. Geom. 68, 23–33 (2000)
De Bruyn, B.: On the universal embedding of the near hexagon related to the extended ternary Golay code. Discrete Math. 312, 883–890 (2012)
De Bruyn, B., De Clerck, F.: On linear representations of near hexagons. Eur. J. Comb. 20, 45–60 (1999)
De Bruyn, B., Shpectorov, S.: Computer code for “The hyperplanes of the near hexagon related to the extended ternary Golay code”. http://cage.ugent.be/geometry/preprints.php
De Bruyn, B., Vanhove, F.: Inequalities for regular near polygons, with applications to \(m\)-ovoids. Eur. J. Comb. 34, 522–538 (2013)
Delsarte, P., Goethals, J.M.: Unrestricted codes with the Golay parameters are unique. Discrete Math. 12, 211–224 (1975)
Frohardt, D., Johnson, P.: Geometric hyperplanes in generalized hexagons of order \((2,2)\). Commun. Algebra 22, 773–797 (1994)
Golay, M.J.E.: Notes on digital coding. Proc. I.R.E 37, 657 (1949)
MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland Mathematical Library, vol. 16. North-Holland Publishing Co., Amsterdam (1977)
Pless, V.: On the uniqueness of the Golay codes. J. Comb. Theory 5, 215–228 (1968)
Pless, V.: More on the uniqueness of the Golay codes. Discrete Math. 106/107, 391–398 (1992)
Ronan, M.A.: Embeddings and hyperplanes of discrete geometries. Eur. J. Comb. 8, 179–185 (1987)
Shult, E.E., Yanushka, A.: Near \(n\)-gons and line systems. Geom. Dedic. 9, 1–72 (1980)
The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.7.5 (2014). http://www.gap-system.org
Wilson, R.A.: The Finite Simple Groups. Graduate Texts in Mathematics, vol. 251. Springer, Berlin (2009)
Yoshiara, S.: Codes associated with some near polygons having three points on each line. Algebra Colloq. 2, 79–96 (1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
De Bruyn, B., Shpectorov, S. The hyperplanes of the near hexagon related to the extended ternary Golay code. Geom Dedicata 202, 9–26 (2019). https://doi.org/10.1007/s10711-018-0400-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-018-0400-z