Designs, Codes and Cryptography

, Volume 43, Issue 1, pp 41–45 | Cite as

Divisible designs with dual translation group

  • Sabine Giese
  • Ralph-Hardo Schulz


Many different divisible designs are already known. Some of them possess remarkable automorphism groups, so called dual translation groups. The existence of such an automorphism group enables us to characterize its associated divisible design as being isomorphic to a substructure of a finite affine space.


Divisible design Dual translation group Affine space 

AMS Classification

05B05 05B30 20B25 51N10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    André J (1954) Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe. Math Z 60:156–186zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Blunck A, Havlicek H, Zanella C (2007) Lifting of divisible designs. Designs, Codes and Cryptography 42:1–14CrossRefMathSciNetGoogle Scholar
  3. 3.
    Cerroni C, Schulz R-H (2000) Divisible designs admitting GL(3, q) as an automorphism group. Geom Dedicata 83:343–350zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cerroni C, Schulz R-H (2001) Divisible designs admitting, as an automorphism group, an orthogonal group or a unitary group. In: Jungnickel D. et al (eds) Finite fields and applications, Springer, Verlag, Berlin, pp. 95–108. Proc. 5th internat. conf. on finite fields and applicationsGoogle Scholar
  5. 5.
    Cerroni C, Spera AG (1999) On divisible designs and twisted field planes. J Comb Des 7:453–464zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Giese S (2005) Block-zerlegbare divisible Designs. URL:, DissertationGoogle Scholar
  7. 7.
    Giese S (2006) Block-decomposability of divisible designs. Submitted to J Comb DesGoogle Scholar
  8. 8.
    Giese S (2006) Constructing block-decomposable divisible designs. PreprintGoogle Scholar
  9. 9.
    Giese S, Havlicek H, Schulz R-H (2005) Some constructions of divisible designs from Laguerre geometries. Discrete Math 301:74–82zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lüneburg H (1980) Translation planes. Springer, New YorkzbMATHGoogle Scholar
  11. 11.
    Schulz R-H (1985) On the classification of translation group-divisible designs. Euro. J. Comb 6:369–374zbMATHMathSciNetGoogle Scholar
  12. 12.
    Schulz R-H (1987) On translation transversal designs with λ > 1. Arch. Math 49:97–102zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Schulz R-H, Spera AG (1998) Construction of divisible designs from translation planes. Eur. J. Comb 19(4):479–486zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Schulz R-H, Spera AG (1998) Divisible designs admitting a Suzuki group as an automorphism group. Boll. Unione Mat Ital 8(1B):705–714MathSciNetGoogle Scholar
  15. 15.
    Spera AG (1992) t-Divisible designs from imprimitive permutation groups. Eur. J. Comb 13:409–417zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Spera AG (2000) Divisible designs associated with translation planes admitting a 2-transitive collineation group on the points at infinity. Aequationes Math 59(1–2):191–200zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Stroppel M (1992) Reconstruction of incidence geometries from groups of automorphisms. Arch. Math 58:621–624zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Freie Universitat BerlinBerlinGermany

Personalised recommendations