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Primitive Polynomials over Finite Fields of Characteristic Two

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Abstract.

Let F q be a finite field of characteristic two. We prove that for any given element a F q , there exists a primitive polynomial of degree n over F q with the m-th (0<m<n) coefficient Σ m =a when n≥7, n is odd.

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References

  1. Chou, W.S., Cohen, S.D.: Primitive elements with zero traces, Dedicated to Professor Ko Chao on the occasion of his 90th birthday. Finite Fields Appl. 7, 125–141 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cohen, S.D.: Primitive elements and polynomials with arbitrary trace. Discrete Math. 83, 1–7 (1990)

    Article  MathSciNet  Google Scholar 

  3. Cohen, S.D.: Primitive elements and polynomials: existence results. Lecture Notes in Pure & Appl. Math. 141, edited by G.L.Mullen and P.J.Shiue, Marcel Dekker, New York, 1993, pp. 43–55

  4. Fan, Sh.Q., Han, W.B.: Character sums over Galois rings and primitive polynomials over finite fields. Finite Fields Appl., S1071-5797(03)00041-8 (in press)

  5. Fan, Sh.Q., Han, W.B.: p-adic formal series and primitive polynomials over finite fields. Proc. Am. Math. Soc. 132(1), 15–31 (2004)

    Article  MATH  Google Scholar 

  6. Fan, Sh.Q., Han, W.B.: p-adic formal series and Cohen’s problem. Glasgow Math. J., scheduled to publish in Jan 2004

  7. Han, W.B.: The coefficients of primitive polynomials over finite fields. Math. Comp. 65(213), 331–340 (1996)

    Article  MATH  Google Scholar 

  8. Han, W.B.: On two exponential sums and their applications. Finite Fields Appl. 3, 115–130 (1997)

    Article  Google Scholar 

  9. Hansen, T., Mullen, G.L.: Primitive polynomials over finite fields, Math. Comp. 59(200), 639–643 (1992), Supplement: S47–S50

    MATH  Google Scholar 

  10. Jungnickel, D., Vanstone, S.A.: On primitive polynomials over finite fields. J. Algebra 124, 337–353 (1989)

    MathSciNet  MATH  Google Scholar 

  11. Kumar, P.V., Helleseth, T., Calderbank, A.R.: An upper bound for Weil exponential sums over Galois rings and applications, IEEE Trans. Inform. Theory. 41(2), 456–468 (1995)

    Article  MATH  Google Scholar 

  12. Li, W.-C.W.: Character sums over p-adic fields. J. Number theor. 74, 181–229 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lidl, R., Niedereiter, H.: Finite Fields. 20 of Encyclopedia of Mathematics and Its Applications. Addison-Wesley, Reading, MA, 1983

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Authors and Affiliations

  1. Department of Applied Mathematics, College of Information Engineering, Information Engineering University, Zhengzhou, 450002, PRC

    Fan Shuqin & Han Wenbao

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  1. Fan Shuqin
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  2. Han Wenbao
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Correspondence to Fan Shuqin.

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Keywords: Finite field, Primitive polynomial, Galois ring, Character sums over Galois ring, The sieve method.

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Shuqin, F., Wenbao, H. Primitive Polynomials over Finite Fields of Characteristic Two. AAECC 14, 381–395 (2004). https://doi.org/10.1007/s00200-003-0140-6

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  • DOI: https://doi.org/10.1007/s00200-003-0140-6

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