Skip to main content
SpringerLink
Log in

The characterization problem for designs with the parameters of AGd(n, q)

  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

We start a new characterization of the geometric 2-design AG d (n,q) among all simple 2-designs with the same parameters by handling the cases d ∈ {1,2,3,n — 2}. For d ≠ 1, our characterization is in terms of line sizes, and for d = 1 in terms of the number of affine hyperplanes. We also show that the number of non-isomorphic resolvable designs with the parameters of AG1(n,q) grows exponentially with linear growth of n.

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. T. Beth, D. Jungniokel and H. Lenz: Design Theory (2nd edition). Cambridge University Press (1999).

    Book  Google Scholar 

  2. F. Buekenhout: Une charactérisation des espaces affines basée sur la notion de droite, Math. Z. 111 1969, 367–371.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Clark, D. Jungniokel and V. D. Tonohev: Correction to: “Exponential bounds on the number of designs with affine parameters”, J. Combin. Des. 19 2011, 156–166.

    Article  MathSciNet  Google Scholar 

  4. C. J. Colbourn and J. H. Dinitz: The Crc Handbook of Combinatorial Designs, Crc Press, oca Raton (1996).

    Book  MATH  Google Scholar 

  5. C. J. Colbourn and J. H. Dinitz: The Crc Handbook of Combinatorial Designs (2nd edition). Chapman & Hall/Crc, Boca Raton (2007).

    MATH  Google Scholar 

  6. P. Dembowski: Eine Kennzeichnung der endlichen affinen Räume, Arch. Math. 15 1964, 146–154.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Dembowski: Berichtigung und Ergänzung zu “Eine Kennzeichnung der endlichen affinen Räume”, Arch. Math. 18 1967, 111–112.

    Article  MathSciNet  MATH  Google Scholar 

  8. P. Dembowski and A. Wagner: Some characterizations of finite projective spaces, Arch. Math. 11 1960, 465–469.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Hall: Automorphisms of Steiner triple systems, Ibm J. Res. Develop. 4 1960, 460–471.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Fuji-Hara and S. A. Vanstone: The existence of orthogonal resolutions of lines in AG(n,q), J. Combin. Theory Ser. 45 1987, 139–147.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. W. P. Hirsohfeld: Projective Geometries over Finite Fields (2nd edition), Oxford University Press (1998).

    Google Scholar 

  12. J. W. P. Hirsohfeld: Finite Projective Spaces of Three Dimensions, Oxford University Press (1985).

    Google Scholar 

  13. D. Jungniokel: Characterizing geometric designs, Rend. Mat. Appl. 30 2010, 111–120.

    MathSciNet  Google Scholar 

  14. D. Jungniokel: Characterizing geometric designs, II, J. Combin. Theory Ser. 118 2011, 623–633.

    Article  MathSciNet  Google Scholar 

  15. D. Jungniokel and V. D. Tonohev: The number of designs with geometric parameters grows exponentially, Des. Codes Cryptogr. 55 2010, 131–140.

    Article  MathSciNet  Google Scholar 

  16. C. Lefèvre-Percsy: Characterizations of designs constructed from affine and projective spaces, European J. Comb. 1 1980, 347–352.

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Lenz: Ein kurzer Weg zur analytischen Geometrie, Math.-Phys. Semesterber. 6 1959, 57–67.

    MathSciNet  MATH  Google Scholar 

  18. K. Metsch: A generalization of a result of Dembowski and Wagner, Des. Codes Cryptogr. 60 2011, 277–282.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Oxenham and R. Casse: On the resolvability of Hall triple systems, Boll. Unione Mat. Ital. Serie 8 1-B (1998), 639–649.

    Google Scholar 

  20. L. Teirlinck: On projective and affine hyperplanes, J. Combin. Theory Ser. 28 1980, 290–306.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. M. Wilson: An existence theory for pairwise balanced designs, Iii. Proof of the existence conjectures, J. Combin. Theor. 18 1975, 71–79.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl für Diskrete Mathematik Optimierung und Operations Research, Universität Augsburg, 86135, Augsburg, Germany

    Dieter Jungnickel

  2. Mathematisches Institut, Universität Gießen, Arndtstrasse 2, 35392, Gießen, Germany

    Klaus Metsch

  3. Department of Mathematics, Ghent University, Krijgslaan 281-S22, 9000, Ghent, Belgium

    Klaus Metsch

Authors
  1. Dieter Jungnickel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Klaus Metsch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dieter Jungnickel.

Additional information

In memoriam Hanfried Lenz

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jungnickel, D., Metsch, K. The characterization problem for designs with the parameters of AGd(n, q). Combinatorica 36, 513–535 (2016). https://doi.org/10.1007/s00493-014-3212-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-014-3212-2

Mathematics Subject Classication (2000)

Search

Navigation