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The Solutions of the Third Power Sum Equation for Niho Type Decimations

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Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3857))

Abstract

We will completely describe the solutions of the equation (x + 1)d = x d + 1 in the field GF(q 2), where q = p k and d is of Niho type, i.e., d ≡ 1 (mod q − 1). Our results have applications in the theory of cross-correlation functions of m-sequences and in the theory of cyclic codes.

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References

  1. Canteaut, A., Charpin, P., Dobbertin, H.: Weight divisibility of cyclic codes, highly nonlinear functions on F2m, and crosscorrelation of maximum-length sequences. SIAM J. Discrete Math. 13(1), 105–138 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Charpin, P.: Cyclic codes with few weights and Niho exponents. J. Combin. Theory Ser. A 108(2), 247–259 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Charpin, P., Tietäväinen, A., Zinoviev, V.A.: On binary cyclic codes with minimum distance three. Problems of Information Transmission 33(4), 287–296 (1997)

    MATH  MathSciNet  Google Scholar 

  4. Dillon, J.F., Dobbertin, H.: New cyclic difference sets with Singer parameters. Finite Fields Appl. 10(3), 342–389 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dobbertin, H.: Almost perfect nonlinear power functions on GF(2n): the Welch case. IEEE Transactions on Information Theory 45(4), 1271–1275 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dobbertin, H., Leander, G., Canteaut, A., Carlet, C., Felke, P., Gaborit, P.: Construction of bent functions via Niho power functions (Submitted)

    Google Scholar 

  7. Helleseth, T.: Some results about the cross-correlation function between two maximal linear sequences. Discrete Mathematics 16(3), 209–232 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  8. Helleseth, T., Lahtonen, J., Rosendahl, P.: On Niho type cross-correlation functions of m-sequences. To appear in Finite Fields and Their Applications

    Google Scholar 

  9. Helleseth, T., Rosendahl, P.: New pairs of m-sequences with four-level cross-correlation. To appear in Finite Fields and Their Applications

    Google Scholar 

  10. Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and Its Applications, vol. 20. Addison-Wesley, Reading (1983)

    MATH  Google Scholar 

  11. Niho, Y.: Multivalued cross-correlation functions between two maximal linear recursive sequences. PhD thesis, University or Southern California (1972)

    Google Scholar 

  12. Rosendahl, P.: Niho type cross-correlation functions and related equations. PhD thesis, University of Turku (2004), Available at http://www.tucs.fi/

  13. van Lint, J.H.: Introduction to coding theory, Graduate Texts in Mathematics, vol. 86. 3rd edn. Springer, Berlin (1999)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Turku, 20014, Turku, Finland

    Kalle Ranto & Petri Rosendahl

Authors
  1. Kalle Ranto
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  2. Petri Rosendahl
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering, University of Hawaii, Honolulu, USA

    Marc P. C. Fossorier

  2. National Institute of Advanced Industrial Science and Technology (AIST), Research Center for Information Security (RCIS),  

    Hideki Imai

  3. Department of Electrical and Computer Engineering, University of California, CA 95616, Davis,  

    Shu Lin

  4. University Paul Sabatier, AAECC/IRIT, 118 route de Narbonne, 31300, Toulouse cedex, France

    Alain Poli

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© 2006 Springer-Verlag Berlin Heidelberg

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Ranto, K., Rosendahl, P. (2006). The Solutions of the Third Power Sum Equation for Niho Type Decimations. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2006. Lecture Notes in Computer Science, vol 3857. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11617983_9

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  • DOI: https://doi.org/10.1007/11617983_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-31423-3

  • Online ISBN: 978-3-540-31424-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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