On the Nth maximum order complexity and the expansion complexity of a Rudin-Shapiro-like sequence

  • Zhimin Sun
  • Xiangyong ZengEmail author
  • Da Lin
Part of the following topical collections:
  1. Special Issue on Sequences and Their Applications


Based on the parity of the number of occurrences of a pattern 10 as a scattered subsequence in the binary representation of integers, a Rudin-Shapiro-like sequence is defined by Lafrance, Rampersad and Yee. The Nth maximum order complexity and the expansion complexity of this Rudin-Shapiro-like sequence are calculated in this paper.


Rudin-Shapiro-like sequence Maximum order complexity Expansion complexity 

Mathematics Subject Classification (2010)

11B50 11B85 11K45 



  1. 1.
    Allouche, J.P., sequences, J. Shallit.: Automatic Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  2. 2.
    Allouche, J.P., Liardet, P.: Generalized Rudin-Shapiro sequences. Acta Arithmet LX.1, 1–27 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Allouche, J.P.: On a Golay-Shapiro-like sequence. Unif. Distrib. Theory 11(2), 205–210 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brillhart, J., Morton, P.: A case study in mathematical research: the Golay-Rudin-Shapiro sequence. Am. Math. Mon. 103(10), 854–869 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chan, L., Grimm, U.: Spectrum of a Rudin-Shapiro-like sequence. Adv. Appl. Math. 87, 16–23 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Diem, C.: On the use of expansion series for stream ciphers. LMS J. Comput. Math. 15, 326–340 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jansen, C.J.A.: Investigations on Nonlinear Streamcipher Systems: Construction and Evaluation Methods. Ph.D.dissertation, Technical University of Delft, Delft (1989)Google Scholar
  8. 8.
    Jansen, C.J.A.: The Maximum Order Complexity of Sequence Ensembles. In: Davies, D.W. (ed.) Advances in Cryptology - EUROCRYPT ’91, Lect. Notes Comput. Sci., vol. 547, pp 153–159. Springer, Berlin (1991)Google Scholar
  9. 9.
    Lafrance, P., Rampersad, N., Yee, R.: Some properties of a Rudin-Shapiro-like sequence. Adv. Appl. Math. 63, 19–40 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mauduit, C., Sárközy, A.: On finite pseudorandom binary sequences II. The Champernowne, Rudin-Shapiro, and Thue-Morse sequences, a further construction. J. Number Theory 73(2), 256–276 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mérai, L., Niederreiter, H., Winterhof, A.: Expansion complexity and linear complexity of sequences over finite fields. Cryptogr. Commun. 9, 501–509 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mérai, L., Winterhof, A.: On the N th linear complexity of automatic sequences. J. Number Theory 187, 415–429 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Müllner, C.: The Rudin-Shapiro sequence and similar sequences are normal along squares. Can. J. Math. 70(5), 1096–1129 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sun, Z., Winterhof, A.: On the maximum order complexity of the Thue-Morse sequence and the Rudin-Shapiro sequence. Preprint (2017)Google Scholar
  15. 15.
    Sun, Z., Winterhof, A.: On the maximum order complexity of subsequences of the Thue-Morse sequence and the Rudin-Shapiro sequence along squares. Int. J. Comput. Math. Comput. Syst. Theory 4(1), 30–36 (2019)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Xing, C.P., Lam, K.Y.: Sequence with almost perfect linear complexity profiles and curves over finite fields. IEEE Trans. Inf. Theory 45(4), 1267–1270 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

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

Authors and Affiliations

  1. 1.Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied MathematicsHubei UniversityWuhanChina

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