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Designs, Codes and Cryptography

, Volume 76, Issue 2, pp 345–360 | Cite as

New pseudo-planar binomials in characteristic two and related schemes

  • Sihuang Hu
  • Shuxing Li
  • Tao Zhang
  • Tao Feng
  • Gennian Ge
Article

Abstract

Planar functions in odd characteristic were introduced by Dembowski and Ostrom in order to construct finite projective planes in 1968. They were also used in the constructions of DES-like iterated ciphers, error-correcting codes, and signal sets. Recently, a new notion of pseudo-planar functions in even characteristic was proposed by Zhou. These new pseudo-planar functions, as an analogue of planar functions in odd characteristic, also bring about finite projective planes. There are three known infinite families of pseudo-planar monomial functions constructed by Schmidt and Zhou, and Scherr and Zieve. In this paper, three new classes of pseudo-planar binomials are provided. Moreover, we find that each pseudo-planar function gives an association scheme which is defined on a Galois ring.

Keywords

Pseudo-planar function Relative difference set Projective plane Association scheme 

Mathematics Subject Classification

Primary 05B10 05E30 94A60 

Notes

Acknowledgments

The authors express their gratitude to the anonymous reviewers for their detailed and constructive comments which are very helpful to the improvement of this paper, and to Prof. Claude Carlet, the Associate Editor, for his excellent editorial job. We would like to thank Professor Qing Xiang and the anonymous reviewer for suggestions on the proofs of Proposition 3.6 and Proposition 3.8, and Dr. Yue Zhou for valuable comments and suggestions. S. Hu was supported by the Scholarship Award for Excellent Doctoral Student granted by Ministry of Education. T. Feng was supported in part by Fundamental Research Fund for the Central Universities of China, Zhejiang Provincial Natural Science Foundation under Grant No.  LQ12A01019, in part by the National Natural Science Foundation of China under Grant No. 11201418, and in part by the Research Fund for Doctoral Programs from the Ministry of Education of China under Grant No. 20120101120089. G. Ge was supported by the National Natural Science Foundation of China under Grant No. 61171198 and Zhejiang Provincial Natural Science Foundation of China under Grant No. LZ13A010001.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Sihuang Hu
    • 1
  • Shuxing Li
    • 1
  • Tao Zhang
    • 1
  • Tao Feng
    • 1
    • 3
  • Gennian Ge
    • 2
    • 3
  1. 1.Department of MathematicsZhejiang UniversityHangzhou China
  2. 2.School of Mathematical SciencesCapital Normal UniversityBeijing China
  3. 3.Beijing Center for Mathematics and Information Interdisciplinary SciencesBeijing China

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