Designs, Codes and Cryptography

, Volume 86, Issue 2, pp 239–250 | Cite as

The order of the automorphism group of a binary \({\varvec{q}}\)-analog of the Fano plane is at most two

  • Michael Kiermaier
  • Sascha Kurz
  • Alfred Wassermann
Part of the following topical collections:
  1. Special Issue on Network Coding and Designs


The question if there exists a q-analog of the Fano plane is open since it was first posed in 1972. For a putative binary q-analog of the Fano plane all automorphisms of order greater than 4 had been excluded previously. Here, it is shown with theoretical and computational methods that the order of the automorphism group of a binary q-analog of the Fano plane is either trivial or of order 2. Moreover, some groups which had been excluded by computer search in Braun et al. (Eur J Comb 51:443–457, 2016) are ruled out as automorphism group by theoretic arguments.


Steiner triple systems q-Analogs of designs Fano plane Automorphism group 

Mathematics Subject Classification

51E20 05B07 05A30 



The authors would like to acknowledge the financial support provided by COST—European Cooperation in Science and Technology—within the Action IC1104 Random Network Coding and Designs over GF(q). The authors also want to thank the IT service center of the University Bayreuth for providing the excellent computing cluster, and especially Dr. Bernhard Winkler for his support. Finally, the authors thank the anonymous referees for pointing out some errors and giving helpful remarks which improved the paper considerably.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthGermany

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