Skip to main content
SpringerLink
Log in

Types of spreads and duality of the parallelisms of PG(3, 5) with automorphisms of order 13

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

A spread is a set of lines of PG(n, q) which partition the point set. A parallelism is a partition of the set of all lines by spreads. Empirical data on parallelisms is of interest both from theoretical point of view, and for different applications. Only 51 explicit examples of parallelisms of PG(3, 5) have been known. We construct all (321) parallelisms of PG(3, 5) with automorphisms of order 13 and classify them by the order of their automorphism group, the number of reguli in their spreads and duality. There are no regular ones among them. There are 19 self-dual parallelisms. We also claim that PG(3, 5) has no point-transitive parallelisms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MATH  Google Scholar 

  2. Baker R.D., Ebert G.L.: Construction of two—dimensional flag—transitive planes. Geom. Dedic. 27, 9–14 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  3. Baker R.D., Ebert G.L.: Regulus-free spreads of \(PG(3, q)\). Des. Codes Cryptogr. 8(1–2), 79–89 (1996).

    MathSciNet  MATH  Google Scholar 

  4. Baker R.D., Ebert G.L.: Two—dimensional flag—transitive planes revisited. Geom. Dedic. 63, 1–15 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  5. Bassalygo L.A., Zinoviev V.A.: Remark on balanced incomplete block designs, near-resolvable block designs, and q-ary constant-weight codes. Probl. Inf. Trans. 53(1), 51–54 (2017).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Beutelspacher A.: On parallelisms in finite projective spaces. Geom. Dedic. 3(1), 35–45 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  8. Braun M.: Construction of a point-cyclic resolution in \(PG(9,2)\). Innov. Incid. Geom. 3, 33–50 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  9. Caliski T., Kageyama S.: On the analysis of experiments in affine resolvable designs. J. Stat. Plan. Inference 138(11), 3350–3356 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  10. Czerwinski T., Oakden D.: The translation planes of order twenty-five. J. Comb. Theory Ser. A 59(2), 193–217 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  11. Denniston R.H.F.: Some packings of projective spaces. Atti Acc. Naz. Lincei Rend. 52(8), 36–40 (1972).

    MathSciNet  MATH  Google Scholar 

  12. Denniston R.H.F.: Cyclic packings of the projective space of order 8. Atti Acc. Naz. Lincei Rend. Cl. Sci. Fix. Mat. Natur. 54, 373–377 (1973).

    MathSciNet  MATH  Google Scholar 

  13. Ding C., Yin J.: A construction of optimal constant composition codes. Des. Codes Cryptogr. 40(2), 157–165 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  14. Etzion T.: Partial k-Parallelisms in finite projective spaces. J. Comb. Des. 23, 101–114 (2015).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Etzion T., Storme L.: Galois geometries and Coding Theory. Des. Codes Cryptogr. 78(1), 311–350 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  17. Etzion T., Vardy A.: Automorphisms of codes in the Grassmann scheme. arXiv:1210.5724 (2012).

  18. GAP: Groups, Algorithms, Programming—A System for Computational Discrete Algebra. https://www.gap-system.org/. Accessed 20 Dec 2017.

  19. Glynn D.: On a set of lines of \(PG(3, q)\) corresponding to a maximal cap contained in the Klein quadric of \(PG(5, q)\). Geom. Dedic. 26(3), 273–280 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  20. Havlicek H., Riesinger R.: Pencilled regular parallelisms. Acta Math. Hung. 153(1), 249–264 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  21. Heinlein D., Honold Th., Kiermaier M., Kurz S.: Generalized vector space partitions. Bayreuth, preprint 2018-2012 p. https://epub.uni-bayreuth.de/3644/ (2018).

  22. Hishida T., Jimbo M.: Possible patterns of cyclic resolutions of the BIB design associated with \(PG(7,2)\). Congr. Numer. 131, 179–186 (1998).

    MathSciNet  MATH  Google Scholar 

  23. Johnson N.L.: Subplane Covered Nets, vol. 222. Monographs and Textbooks in Pure and Applied MathematicsMarcel Dekker, New York (2000).

    Book  MATH  Google Scholar 

  24. Johnson N.L.: Some new classes of finite parallelisms. Note Mat. 20(2), 77–88 (2000/2001).

  25. Johnson N.L.: Parallelisms of projective spaces. J. Geom. 76(1–2), 110–182 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  26. Johnson N.L.: Combinatorics of Spreads and Parallelisms. CRC Press, New York (2010).

    Book  MATH  Google Scholar 

  27. Johnson N.L., Montinaro A.: The transitive t-parallelisms of a finite projective space. Adv. Geom. 12(3), 401–429 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  28. Johnson S., Weller S.: Resolvable 2-designs for regular low density parity-check codes. IEEE Trans. Commun. 51(9), 1413–1419 (2003).

    Article  Google Scholar 

  29. Kurosawa K., Kageyama S.: New bound for affine resolvable designs and its application to authentication codes. In: Du D.Z., Li M. (eds.) Computing and Combinatorics. COCOON 1995. Lecture Notes in Computer Science, vol 959. Springer, Berlin, Heidelberg (1995).

  30. Koetter R., Kschischang F.R.: Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theory 54, 3579–3591 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  31. Lunardon G.: On regular parallelisms in \(PG(3, q)\). Discret. Math. 51, 229–335 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  32. Olmez O., Ramamoorthy A.: Fractional repetition codes with flexible repair from combinatorial designs. IEEE Trans. Inf. Theory 62(4), 1565–1591 (2016).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Prince A.R.: Parallelisms of \(PG(3,3)\) invariant under a collineation of order 5. In: Johnson N.L. (eds.) Mostly Finite Geometries. Lecture Notes Pure Applied Mathematics, vol 190, pp. 383–390. Marcel Dekker, New York (l997).

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Ruj S., Seberry J., Roy B.: Key predistribution schemes using block designs in wireless sensor networks. In: 12-th IEEE International Conference on Computational Science and Engineering (CSE), pp. 873–878 (2009).

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Semakov N., Zinovev V.: Equidistant q-ary codes with maximal distance and resolvable balanced incomplete block designs. Probl. Peredachi Inf. 4(2), 3–10 (1968).

    MathSciNet  MATH  Google Scholar 

  39. Stinson D.R.: Combinatorial Designs: Constructions and Analysis. Springer, New York (2004).

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Storme L.: Finite Geometry. In: Colbourn C.J., Dinitz J.H. (eds.) The CRC Handbook of Combinatorial Designs, 2nd edn, pp. 702–729. CRC Press, Boca Raton (2006).

    Google Scholar 

  42. Topalova S., Zhelezova S.: On transitive parallelisms of \(PG(3,4)\). Appl. Algebra Eng. Commun. Comput. 24(3–4), 159–164 (2013).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Topalova S., Zhelezova S.: New regular parallelisms of \(PG(3,5)\). J. Comb. Des. 24(10), 473–482 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  45. White C.: Two cyclic arrangement problems in finite projective geometry: parallelisms and two-intersection set, PhD thesis, California Institute of Technology (2002).

  46. Walker M.: Spreads covered by derivable partial spreads. J. Comb. Theory Ser. A 38(2), 113–130 (1985).

    Article  MathSciNet  MATH  Google Scholar 

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

  48. Zhelezova S.: Cyclic parallelisms of \(PG(5,2)\). Math. Balkanica 24(1–2), 141–146 (2010).

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful to professor A. Betten from Colorado State University for focusing our attention to invariant 2, and to the anonymous referees whose remarks contributed to a better presentation and motivation of the results.

Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics, BAS, Sofia, Bulgaria

    Svetlana Topalova & Stela Zhelezova

Authors
  1. Svetlana Topalova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stela Zhelezova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Svetlana Topalova.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

Part of the results were announced at the Seventh International Workshop on Optimal Codes and Related Topics, September, 2013, Albena, Bulgaria.

This work is supported by the Bulgarian National Science Fund [Contract No. DH 02/2, 13.12.2016].

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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, 495–507 (2019). https://doi.org/10.1007/s10623-018-0558-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-018-0558-2

Keywords

Mathematics Subject Classification

Search

Navigation