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Algebraic Codes and Geometry of Some Classical Generalized Polygons

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Groups of Exceptional Type, Coxeter Groups and Related Geometries

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 82))

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Abstract

Some results about the geometry of, and the \(q\)-ary codes associated with the finite generalized polygons \(W(q), q=2^{n}\); \(H(q)\), \(q=3^{n}\); and \(\mathcal {O}(q)\) , \(q=2^{2m+1}\), are surveyed to put in context a few simple observations and state a few related questions. In Sect. 4, using the polarity in the case of \( W(q)\), \(q=2^{2m+1}\), and that of \(H(q)\), \(q=3^{2m+1}\), we present a nondegenerate symmetric bilinear form on, and a polarity of, the \(q\)-ary code associated with each of these geometries which is stabilized by the centralizer of the polarity in the automorphism group of the geometry.

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Acknowledgments

I thank H. Van Maldeghem for several very useful remarks and his permission to include Proposition 2.12 (iii) here Ilaria Cardinali for discussions on the material in Sect 2B(i. a) and T. Penttila for some useful remarks. Finally, I dedicate this article to the memory of two recently departed, very important maternal persons in my life Kanakammagaru and Venkatalakshammagaru.

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  1. Indian Statistical Institute Division of Theoretical Statistics and Mathematics, 8th Mile, Mysore Road, R.V. College Post, Bangalore, India

    N. S. Narasimha Sastry

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  1. Stat Math Unit, Indian Statistical Institute, Bangalore, Karnataka, India

    N.S. Narasimha Sastry

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Narasimha Sastry, N. (2014). Algebraic Codes and Geometry of Some Classical Generalized Polygons. In: Sastry, N. (eds) Groups of Exceptional Type, Coxeter Groups and Related Geometries. Springer Proceedings in Mathematics & Statistics, vol 82. Springer, New Delhi. https://doi.org/10.1007/978-81-322-1814-2_14

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