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On four codes with automorphism group \(P\Sigma L(3,4)\) and pseudo-embeddings of the large Witt designs

  • Bart De BruynEmail author
  • Mou Gao
Article
  • 11 Downloads

Abstract

A pseudo-embedding of a point-line geometry is a representation of the geometry into a projective space over the field \({\mathbb F}_2\) such that every line corresponds to a frame of a subspace. Such a representation is called homogeneous if every automorphism of the geometry lifts to an automorphism of the projective space. In this paper, we determine all homogeneous pseudo-embeddings of the three Witt designs that arise by extending the projective plane \({\mathrm{PG}}(2,4)\). Along our way, we come across some codes with automorphism group \(P\Sigma L(3,4)\) and sets of points of \({\mathrm{PG}}(2,4)\) that have a particular intersection pattern with Baer subplanes or hyperovals.

Keywords

Witt design Mathieu group (homogeneous) Pseudo-embedding Even set Linear code Hyperoval Baer subplane 

Mathematics Subject Classification

51E20 94B05 05B05 51A45 20C20 

Notes

Acknowledgements

The author Mou Gao, is supported by the State Scholarship Fund (File No. 201806065052) and the National Natural Science Foundation of China (Grant No. 71771035).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics: Algebra and GeometryGhent UniversityGentBelgium

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