Skip to main content
Log in

Naive configurations

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

Abstract

We describe a new way to construct finite geometric objects. For every \(k\) we obtain a symmetric configuration \(\mathcal{E }(k-1)\) with \(k\) points on a line. In particular, we have a constructive existence proof for such configurations. The method is very simple and purely geometric. It also produces interesting periodic matrices.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Brualdi, R.A., Pless, V.S.: Greedy codes. J. Comb. Theory, Series A 64, 10–30 (1993)

  2. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to algorithmsm, chap. 16. In: Greedy Algorithms. MIT Press, Cambridge (2001)

  3. Conway, J.H.: Integral lexicographic Codes. Discret. Math. 83, 219–235 (1990)

    Google Scholar 

  4. Conway, J.H., Sloane, N.J.A.: Lexicographic codes: error-correcting codes from game theory. IEEE Trans. Inf. Theory IT-32(3), 337–348 (1986)

    Google Scholar 

  5. Dembowski, P.: Finite Geometries. Ergebnisse der Mathematik und ihrer Grenzgebiete 44, Springer-Verlag, Berlin, Heidelberg, New York (1968)

  6. Edgar, T.: First-Best Projective Planes and Related Structures. Diplomarbeit, Tübingen (2009)

  7. Euler, L.: Recherches sur une nouvelle espèce des quarrés magiques. Verh. Zeeuwsch. Genootsch. Wetensch. Vlissingen 9, 85–239 (1782)

    Google Scholar 

  8. Floyd, R.W.: Non-deterministic algorithms. J. ACM 14, 636–644 (1967)

    Google Scholar 

  9. Gropp, H.: Configurations and their realization. Discret. Math. 174, 137–151 (1997)

    Google Scholar 

  10. Hering, C.H., Krebs, A.: A partial plane of order 6 constructed from the icosahedron. Des. Codes Cryptogr. 44, 287–292 (2007)

    Google Scholar 

  11. Krebs, A., Hering, C.H., Edgar, T.: First choice constructions for non-symmetric configurations (to appear)

  12. Lam, C.W.H.: The search for a finite projective plane of order 10. Am. Math. Mon. 98, 305–318 (1991)

    Google Scholar 

  13. ProjFinder: http://www.mathematik.uni-tuebingen.de/ab/gruppen/hering/main.html. Accessed 7 Feb 2013

  14. MacWilliams, F.C., Sloane, N.J.A., Thompson, J.G.: On the existence of a projective plane of order 10. J. Comb. Theory 14 A, 66–78 (1973)

    Google Scholar 

  15. Ward, H.N.: A selection of divisible lexicographic codes. Int. J. Inf. Coding Theory 1(4), 410–428 (2010)

    Google Scholar 

Download references

Acknowledgments

Authors are grateful to the reviewers for many very helpful comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Christoph Hering.

Additional information

Communicated by J. D. Key.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hering, C., Krebs, A. & Edgar, T. Naive configurations. Des. Codes Cryptogr. 72, 719–731 (2014). https://doi.org/10.1007/s10623-013-9797-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-013-9797-4

Keywords

Mathematics Subject Classification (2010)

Navigation