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Improved Bounds for the Number of Occurrences of Elements in Linear Recurrence Sequences Over Galois Rings

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Abstract

We establish bounds for the number of occurrences of elements on segments of linear recurrence sequences of vectors over Galois rings. We use the method of exponential sums for this problem. We improve known results with the help of a new class of exponential sums.

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References

  1. H. Hu, D. Feng, and W. Wu, “Incomplete exponential sums over Galois rings with applications to some binary sequences derived from \( {{\mathbb{Z}}_{{{2^l}}}} \),” IEEE Trans. Inform. Theory, 52, No. 5, 2260–2265 (2006).

    Article  MathSciNet  Google Scholar 

  2. O. V. Kamlovskii, “Estimates for the frequencies of the occurrence of zeros in linear recurrent sequences of vectors,” Chebyshevskii Sb., 6, No. 1, 135–146 (2005).

    MATH  MathSciNet  Google Scholar 

  3. O. V. Kamlovskii, “Frequency characteristics of linear recurrence sequences over Galois rings,” Mat. Sb., 200, No. 4, 31–52 (2009).

    Article  MathSciNet  Google Scholar 

  4. O. V. Kamlovskii, “The Sidelnikov method for estimating the number of signs on segments of linear recurrence sequences over Galois rings,” Mat. Zametki, 91, No. 3, 371–382 (2012).

    Article  Google Scholar 

  5. O. V. Kamlovskii and A. S. Kuzmin, “Bounds for the number of occurrences of elements in a linear recurring sequence over a Galois ring,” Fundam. Prikl. Mat., 6, No. 4, 1083–1094 (2000).

    MathSciNet  Google Scholar 

  6. D. E. Knuth, The Art of Computer Programming, Vol. 2 [Russian translation], Vil’yams, Moscow (2000).

  7. N. M. Korobov, “The distribution of nonresidues and primitive roots in recurrence series,” Dokl. Akad. Nauk SSSR, 8, No. 4, 603–606 (1953).

    MathSciNet  Google Scholar 

  8. V. L. Kurakin, A. S. Kuzmin, A. V. Mikhalev, and A. A. Nechaev, “Linear recurring sequences over rings and modules,” J. Math. Sci., 76, No. 6, 2793–2915 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  9. A. S. Kuzmin, V. L. Kurakin, and A. A. Nechaev, “Pseudorandom and polylinear sequences,” in: Works in Discrete Mathematics, Vol. 1 [in Russian], TVP, Moscow (1997), pp. 139–202.

  10. B. R. McDonald, Finite Rings with Identity, Marcel Dekker, New York (1974).

    MATH  Google Scholar 

  11. D. A. Mikhailov, “Unitary polylinear registers and its periods,” Diskret. Mat., 14, No. 1, 30–59 (2002).

    Article  Google Scholar 

  12. A. A. Nechaev, “Kerdoc code in a cyclic form,” Diskret. Mat., 1, No. 4, 123–139 (1989).

    MATH  MathSciNet  Google Scholar 

  13. V. I. Nechaev, “Distribution of signs in a sequence of rectangular matrices over a finite fields,” Tr. Mat. Inst. Steklova, 218, 335–342 (1997).

    MathSciNet  Google Scholar 

  14. H. Niederreiter, “Distribution properties of feedback shift register sequences,” Probl. Control Inform. Theory, 15, No. 1, 19–34 (1986).

    MATH  MathSciNet  Google Scholar 

  15. A. G. Shanbhag, P. V. Kumar, and T. Helleseth, “Upper bound for a hybrid sum over Galois rings with applications to aperiodic correlation of some q-ary sequences,” IEEE Trans. Inform. Theory, 42, No. 1, 250–254 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  16. I. E. Shparlinskii, “On distribution of values of recurrence sequences,” Probl. Peredachi Inform., 25, No. 2, 46–53 (1989).

    MathSciNet  Google Scholar 

  17. V. M. Sidelnikov, “Estimates for the number of appearances of signs on a segment of a recurrence sequence over a finite fields,” Diskret. Mat., 3, No. 2, 87–95 (1991).

    Google Scholar 

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Authors and Affiliations

  1. Certification Research Center, Moscow, Russia

    O. V. Kamlovskii

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  1. O. V. Kamlovskii
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Correspondence to O. V. Kamlovskii.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 7, pp. 97–115, 2011/12.

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Kamlovskii, O.V. Improved Bounds for the Number of Occurrences of Elements in Linear Recurrence Sequences Over Galois Rings. J Math Sci 197, 512–524 (2014). https://doi.org/10.1007/s10958-014-1731-4

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  • DOI: https://doi.org/10.1007/s10958-014-1731-4

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