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Parallelisms of \(\mathrm{PG}(3,4)\) Invariant Under Cyclic Groups of Order 4

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Algebraic Informatics (CAI 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11545))

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Abstract

A spread in \(\mathrm{PG}(n,q)\) is a set of mutually skew lines which partition the point set. A parallelism is a partition of the set of lines by spreads. The classification of parallelisms in small finite projective spaces is of interest for problems from projective geometry, design theory, network coding, error-correcting codes, cryptography, etc. All parallelisms of \(\mathrm{PG}(3,2)\) and \(\mathrm{PG}(3,3)\) are known and parallelisms of \(\mathrm{PG}(3,4)\) which are invariant under automorphisms of odd prime orders and under the Baer involution have already been classified. In the present paper, we classify all (we establish that their number is 252738) parallelisms in \(\mathrm{PG}(3,4)\) that are invariant under cyclic automorphism groups of order 4. We compute the order of their automorphism groups and obtain invariants based on the type of their spreads and duality.

The research of the second and the third author is partially supported by the Bulgarian National Science Fund under Contract No. DH 02/2, 13.12.2016.

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Author information

Authors and Affiliations

  1. Colorado State University, Fort Collins, USA

    Anton Betten

  2. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Svetlana Topalova & Stela Zhelezova

Authors
  1. Anton Betten
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  2. Svetlana Topalova
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  3. Stela Zhelezova
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Corresponding author

Correspondence to Stela Zhelezova .

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Editors and Affiliations

  1. University of Niš, Niš, Serbia

    Miroslav Ćirić

  2. University of Leipzig , Leipzig, Germany

    Manfred Droste

  3. Université Paris Denis Diderot and CNRS, Paris, France

    Jean-Éric Pin

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Betten, A., Topalova, S., Zhelezova, S. (2019). Parallelisms of \(\mathrm{PG}(3,4)\) Invariant Under Cyclic Groups of Order 4. In: Ćirić, M., Droste, M., Pin, JÉ. (eds) Algebraic Informatics. CAI 2019. Lecture Notes in Computer Science(), vol 11545. Springer, Cham. https://doi.org/10.1007/978-3-030-21363-3_8

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  • DOI: https://doi.org/10.1007/978-3-030-21363-3_8

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  • Print ISBN: 978-3-030-21362-6

  • Online ISBN: 978-3-030-21363-3

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