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Strongly regular graphs arising from non-weakly regular bent functions

  • Ferruh ÖzbudakEmail author
  • Rumİ Melİh Pelen
Article
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Part of the following topical collections:
  1. Special Issue on Boolean Functions and Their Applications

Abstract

In this paper, we study two special subsets of a finite field of odd characteristics associated with non-weakly regular bent functions. We show that those subsets associated to non-weakly regular even bent functions in the GMMF class (see Çesmelioğlu et al. Finite Fields Appl. 24, 105–117 2013) are never partial difference sets (PDSs), and are PDSs if and only if they are trivial subsets. Moreover, we analyze the two known sporadic examples of non-weakly regular ternary bent functions given in Helleseth and Kholosha (IEEE Trans. Inf. Theory 52(5), 2018–2032 2006, Cryptogr. Commun. 3(4), 281–291 2011). We observe that corresponding subsets are non-trivial partial difference sets. We show that they are the union of some cyclotomic cosets and so correspond to 2-class fusion schemes of a cyclotomic scheme. We also present a further construction giving non-trivial PDSs from certain p-ary functions which are not bent functions.

Keywords

Non weakly regular bent functions Partial difference sets Cyclotomic association schemes Fusion schemes Strongly regular graphs 

Mathematics Subject Classification (2010)

11T22 11T71 05E30 

Notes

Acknowledgments

The authors extend thanks to the anonymous reviewers for their valuable comments and suggestions, which improved the quality and presentation of the manuscript.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  2. 2.Department of MathematicsMiddle East Technical UniversityAnkaraTurkey
  3. 3.Middle Black Sea Devolopment AgencyTekkeköyTurkey

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