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A note on almost difference sets in nonabelian groups

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Abstract

We present a new construction of almost difference sets. The construction occurs in nonabelian groups of order 4N with a subgroup H of order N so that H has an (N,\(\frac{N-1}{2}\),\(\frac{N-3}{4}\)) difference set (and hence N must be an integer that is 3 (mod 4)).

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References

  1. Arasu K.T., Ding C., Helleseth T., Kumar P.V., Martinsen H.: Almost difference sets and their sequences with optimal autocorrelation. IEEE Trans. Inform. Theory 47, 2834–2943 (2001).

    MathSciNet  MATH  Google Scholar 

  2. Carlet C., Ding C., Yuan J.: Linear codes from perfect nonlinear mappings and their secret sharing schemes. IEEE Trans. Inform. Theory 51, 2089–2102 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  3. Davis J.: Almost difference sets and reversible difference sets. Arch. Math. 59, 595–602 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  4. Ding C.: The differential cryptanalysis and design of the natural stream ciphers. Lect. Note Comp. Sci. 809, 101-115, Springer-Verlag (1994).

  5. Ding C.: Codes from Difference Sets. World Scientific, Hackensack (2015).

    Google Scholar 

  6. Ding C., Helleseth T., Martinsen H.M.: New families of binary sequences with optimal three-level autocorrelation. IEEE Trans. Inform. Theory 47, 428–433 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  7. Ding C., Pott A., Wang Q.: Constructions of almost difference sets from finite fields. Des. Codes Cryptogr. 72, 581–592 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  8. Nowak K.: A survey on almost difference sets (2014). arXiv:1409.0114v1

  9. Smith K.: Non-Abelian Hadamard difference sets. J. Comb. Theory 70, 144–156 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  10. Tang X., Ding C.: New classes of balanced quaternary and almost balanced binary sequences with optimal autocorrelation value. IEEE Trans. Inform. Theory 56(12), 6398–6405 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhang Y., Lei J.G., Zhang S.P.: A new family of almost difference sets and some necessary conditions. IEEE Trans. Inform. Theory 52(5), 2052–2061 (2006).

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

David Clayton is an undergraduate at the University of Richmond. He was supported in the summer of 2016 by a University of Richmond summer research fellowship. We thank Dr. James A. Davis at the University of Richmond for serving as advisor on this project.

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  1. University of Richmond, Richmond, VA, 23173, USA

    David Clayton

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  1. David Clayton
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Correspondence to David Clayton.

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Communicated by C. Ding.

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Clayton, D. A note on almost difference sets in nonabelian groups. Des. Codes Cryptogr. 86, 1405–1410 (2018). https://doi.org/10.1007/s10623-017-0403-z

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  • DOI: https://doi.org/10.1007/s10623-017-0403-z

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