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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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References

  1. Baker, R.D.: Partitioning the planes of \(AG_{2m}(2)\) into 2-designs. Discret. Math. 15, 205–211 (1976)

    Article  MathSciNet  Google Scholar 

  2. Bamberg, J.: There are no regular packings of \({\rm PG}(3,3)\) or \({\rm PG}(3,4)\). https://symomega.wordpress.com/2012/12/01/. Accessed 24 Jan 2019

  3. Betten, A.: The packings of \({\rm PG}(3,3)\). Des. Codes Cryptogr. 79(3), 583–595 (2016)

    Article  MathSciNet  Google Scholar 

  4. Betten, A.: Orbiter - a program to classify discrete objects, 2016–2018. https://github.com/abetten/orbiter. Accessed 24 Jan 2019

  5. Betten, A.: Spreads and packings - an update. In: Booklet of Combinatorics 2016, Maratea (PZ), Italy, 29th May–5th June, p. 45 (2016)

    Google Scholar 

  6. Betten, A., Topalova, S., Zhelezova, S.: Parallelisms of \({\rm PG}(3,4)\) invariant under a baer involution. In: Proceedings of the 16th International Workshop on Algebraic and Combinatorial Coding Theory, Svetlogorsk, Russia, pp. 57–61 (2018). http://acct2018.skoltech.ru/

  7. Beutelspacher, A.: On parallelisms in finite projective spaces. Geom. Dedicata 3(1), 35–40 (1974)

    Article  MathSciNet  Google Scholar 

  8. Colbourn, C., Dinitz, J. (eds.) Handbook of combinatorial designs. In: Rosen, K. (ed.) Discrete Mathematics and its Applications, 2nd edn. CRC Press, Boca Raton (2007)

    Google Scholar 

  9. Dempwolff, U., Reifart, A.: The classification of the translation planes of order \(16\). I. Geom. Dedicata 15(2), 137–153 (1983)

    MathSciNet  MATH  Google Scholar 

  10. Denniston, R.H.E.: Some packings of projective spaces. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 52(8), 36–40 (1972)

    MathSciNet  MATH  Google Scholar 

  11. Eisfeld, J., Storme L.: (Partial) t-spreads and minimal t-covers in finite projective spaces. Lecture notes from the Socrates Intensive Course on Finite Geometry and its Applications, Ghent, April 2000

    Google Scholar 

  12. Etzion, T., Silberstein, N.: Codes and designs related to lifted MRD codes. IEEE Trans. Inform. Theory 59(2), 1004–1017 (2013)

    Article  MathSciNet  Google Scholar 

  13. GAP - Groups, algorithms, programming - a system for computational discrete algebra. http://www.gap-system.org/. Accessed 24 Jan 2019

  14. Gruner, A., Huber, M.: New combinatorial construction techniques for low-density parity-check codes and systematic repeat-accumulate codes. IEEE Trans. Commun. 60(9), 2387–2395 (2012)

    Article  Google Scholar 

  15. Fuji-Hara, R.: Mutually 2-orthogonal resolutions of finite projective space. Ars Combin. 21, 163–166 (1986)

    MathSciNet  MATH  Google Scholar 

  16. Johnson, N.L.: Some new classes of finite parallelisms. Note Mat. 20(2), 77–88 (2000)

    MathSciNet  MATH  Google Scholar 

  17. Johnson, N.L.: Combinatorics of Spreads and Parallelisms. Chapman & Hall Pure and Applied Mathematics. CRC Press, Boca Raton (2010)

    Book  Google Scholar 

  18. Magma. The Computational Algebra Group within the School of Mathematics and Statistics of the University of Sydney (2004)

    Google Scholar 

  19. McKay, B.: Nauty User’s Guide (Version 2.4). Australian National University (2009)

    Google Scholar 

  20. Penttila, T., Williams, B.: Regular packings of \({\rm PG}(3,q)\). Eur. J. Comb. 19(6), 713–720 (1998)

    Article  MathSciNet  Google Scholar 

  21. Prince, A.R.: Parallelisms of \(PG(3,3)\) invariant under a collineation of order 5. In: Johnson, N.L. (ed.) Mostly Finite Geometries. Lecture Notes in Pure and Applied Mathematics, vol. 190, pp. 383–390. Marcel Dekker, New York (1997)

    Google Scholar 

  22. Prince, A.R.: Uniform parallelisms of PG(3,3). In: Hirschfeld, J., Magliveras, S., Resmini, M. (eds.) Geometry, Combinatorial Designs and Related Structures. London Mathematical Society Lecture Note Series, vol. 245, pp. 193–200. Cambridge University Press, Cambridge (1997)

    Chapter  Google Scholar 

  23. Prince, A.R.: The cyclic parallelisms of \(PG(3,5)\). Eur. J. Comb. 19(5), 613–616 (1998)

    Article  MathSciNet  Google Scholar 

  24. Sarmiento, J.: Resolutions of \(PG(5,2)\) with point-cyclic automorphism group. J. Comb. Des. 8(1), 2–14 (2000)

    Article  MathSciNet  Google Scholar 

  25. Stinson, D.R.: Combinatorial Designs: Constructions and Analysis. Springer, New York (2004). https://doi.org/10.1007/b97564

    Book  MATH  Google Scholar 

  26. Stinson, D.R., Vanstone, S.A.: Orthogonal packings in \(PG(5,2)\). Aequationes Math. 31(1), 159–168 (1986)

    Article  MathSciNet  Google Scholar 

  27. Storme, L.: Finite geometry. In: Colbourn, C., Dinitz, J., Handbook of Combinatorial Designs, Rosen, K. (eds.) Discrete Mathematics and its Applications, 2nd edn, pp. 702–729. CRC Press, Boca Raton (2007)

    Google Scholar 

  28. Topalova, S., Zhelezova, S.: On transitive parallelisms of \({\rm PG}(3,4)\). Appl. Algebra Engrg. Comm. Comput. 24(3–4), 159–164 (2013)

    Article  MathSciNet  Google Scholar 

  29. Topalova, S., Zhelezova, S.: On point-transitive and transitive deficiency one parallelisms of \({\rm PG}(3,4)\). Des. Codes Cryptogr. 75(1), 9–19 (2015)

    Article  MathSciNet  Google Scholar 

  30. Topalova, S., Zhelezova, S.: New regular parallelisms of \(PG(3,5)\). J. Comb. Des. 24, 473–482 (2016)

    Article  MathSciNet  Google Scholar 

  31. Topalova, S., Zhelezova, S.: New parallelisms of \({\rm PG}(3,4)\). Electron. Notes Discret. Math. 57, 193–198 (2017)

    Article  Google Scholar 

  32. Topalova, S., Zhelezova, S.: Types of spreads and duality of the parallelisms of \(PG(3,5)\) with automorphisms of order \(13\). Des. Codes Cryptogr. 87(2–3), 495–507 (2019)

    Article  MathSciNet  Google Scholar 

  33. Zaicev, G., Zinoviev, V., Semakov, N.: Interrelation of Preparata and Hamming codes and extension of Hamming codes to new double-error-correcting codes. In: 1971 Proceedings of Second International Symposium on Information Theory, Armenia, USSR, pp. 257–263. Academiai Kiado, Budapest (1973)

    Google Scholar 

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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 .

Editor information

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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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-21362-6

  • Online ISBN: 978-3-030-21363-3

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