Designs, Codes and Cryptography

, Volume 75, Issue 3, pp 405–427 | Cite as

Self-embeddings of Hamming Steiner triple systems of small order and APN permutations

  • Josep Rifà
  • Faina I. Solov’eva
  • Mercè Villanueva
Article

Abstract

The classification, up to isomorphism, of all self-embedding monomial power permutations of Hamming Steiner triple systems of order \(n=2^m-1\) for small \(m\,(m \le 22)\), is given. As far as we know, for \(m\! \in \! \{5,7,11,13,17,19 \}\), all given self-embeddings in closed surfaces are new. Moreover, they are cyclic for all \(m\) and nonorientable at least for all \(m \le 19\). For any non prime \(m\), the nonexistence of such self-embeddings in a closed surface is proven. The rotation line spectrum for self-embeddings of Hamming Steiner triple systems in pseudosurfaces with pinch points as an invariant to distinguish APN permutations or, in general, to classify permutations, is also proposed. This invariant applied to APN monomial power permutations gives a classification which coincides with the classification of such permutations via CCZ-equivalence, at least up to \(m\le 17\).

Keywords

APN functions Hamming codes Self-embeddings Steiner triple systems 

Mathematics Subject Classification

94B15 94A60 

Notes

Acknowledgments

This work was partially supported by the Spanish MICINN under Grants MTM2009-08435 and TIN2010-17358, and by the Catalan AGAUR under Grant 2009SGR1224. The work of the second author was supported by Grants RFBR 12-01-00631-a and NSh-1939.2014.1 of President of Russia for Leading Scientific Schools.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Josep Rifà
    • 1
  • Faina I. Solov’eva
    • 2
  • Mercè Villanueva
    • 1
  1. 1.Department of Information and Communications EngineeringUniversitat Autònoma de BarcelonaBellaterraSpain
  2. 2.Sobolev Institute of Mathematics and Novosibirsk State UniversityNovosibirskRussia

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