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.
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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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Appendix
Appendix
When \(n\) is odd, the second eigenmatrix of the association scheme is
When \(n\) is even, the second eigenmatrix of the association scheme is
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Hu, S., Li, S., Zhang, T. et al. New pseudo-planar binomials in characteristic two and related schemes. Des. Codes Cryptogr. 76, 345–360 (2015). https://doi.org/10.1007/s10623-014-9958-0
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DOI: https://doi.org/10.1007/s10623-014-9958-0