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Fast Computations of Gröbner Bases and Blind Recognitions of Convolutional Codes

  • Conference paper
Arithmetic of Finite Fields (WAIFI 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4547))

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Abstract

This paper provides a fast algorithm for Gröbner bases of homogenous ideals of the ring \(\Bbb{F}[x,y]\) over a field \(\Bbb{F}\). The computational complexity of the algorithm is O(N 2), where N is the maximum degree of the input generating polynomials. The new algorithm can be used to solve a problem of blind recognition of convolutional codes. This is a new generalization of the important problem of synthesis of a linear recurring sequence.

This work was supported by the National Natural Science Foundation of China (60673082,90204013),and Special Funds of Authors of Excellent Doctoral Dissertation in China(200084).

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References

  • Althaler, J., Dür, A.: Finite Linear Recurring Sequences and Homogeneous Ideals. AAECC 7, 377–390 (1996)

    MATH  Google Scholar 

  • Althaler, J., Dür, A.: A Generalization of the Massey-Ding Algorithm. AAECC 9, 1–14 (1998)

    Article  MATH  Google Scholar 

  • Berlekamp, E.R.: Algebraic Coding Theory. McGrw-Hill, New York (1968)

    MATH  Google Scholar 

  • Buchberger, B., Gröbner, B.: An Algorithmic Method in Polynomial Ideal Theory. In: Bose, N.K. (ed.) Multidimensional Systems Theory (1984)

    Google Scholar 

  • Cheng, M.H.: Generalised Berlekamp-Massey Algorithm. IEE Proc.Comm. 149(4), 207–210 (2002)

    Article  Google Scholar 

  • Faugére, J.C., Gianni, P., Lazard, D., Mora, T.: Efficient Computation of Zero-dimensional Gröbner bases by Change of Ordering. J.Symb. Comp. 16, 329–344 (1993)

    Article  MATH  Google Scholar 

  • Faugére, J.: A New Efficient Algorithm for Computing Gröbner Bases (F4). J. Pure and Applied Algebra 139, 61–83 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Faugére, J.: A New Efficient Algorithm for Computing Gröbner Bases Without Reduction to Zero (F5). In: Proc. of ISSAC 02, pp. 75–83. ACM Press, New York (2002)

    Chapter  Google Scholar 

  • Feng, K.L.: A Generation of the Berlekamp-Massey Algorithm for Mutisequence Shift-register Synthesis with Applications to Decoding Cyclic Codes. IEEE Trans. on Inform. Theory 37, 1274–1287 (1991)

    Article  MATH  Google Scholar 

  • Fitzpatrick, P.: Solving a Multivariable Congruence by Change of Term Order. J. Symb.Comp. 11, 1–15 (1997)

    Google Scholar 

  • Golic, J.D.: Vectorial Boolean Functions and Induced Algebraic Equations. IEEE Trans. on Inform. Theory 52(2), 528–537 (2006)

    Article  MathSciNet  Google Scholar 

  • Heydtmann, A.E., Jensen, J.M.: On the Equivalence of the Berlekamp-Massey and the Euclidean Algorithms for Decoding. IEEE Trans. Inform.Theory 46(7), 2614–2624 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Kailath, T.: Encounters with the Berlekamp-Massey Algorithm. In: Blahut, R.E., et al. (ed.) Communication and Cryptography, pp. 209–220. Kluwer Academic Publisher, Boston, MA (1994)

    Google Scholar 

  • Kuijper, M., Willems, J.C.: On Constructing a Shortest Linear Recurrence Relation. IEEE Trans. Automat. Contr. 42(11), 1554–1558 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • Lazard, D.: A Note on Upper Bounds for Ideal-Theoretic Problems. J. Symbolic Computation 13(3), 231–233 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  • Lu, P.Z., Liu, M.L.: Gröbner Basis of Characteristic Ideal of LRS over UFD, Science in China (Series A), vol. 28(6) (1998)

    Google Scholar 

  • Lu, P.Z.: Synthesis of Sequences and Efficient Decoding for a Class of Algebraic Geometry Codes. Acta. Electronic Sinica 21(1), 1–10 (1993)

    Google Scholar 

  • Lu, P.Z.: Structure of Groebner Bases for the Characteristic Ideal of a Finite Linear Recursive Sequence, AAECC-13, HI,USA (November 14-November 19, 1999)

    Google Scholar 

  • Lu, P.Z., Shen, L., Zou, L., Luo, X.Y.: Blind Recognition of Punctured Concolutional Codes. Science in China (Series F) 48(4), 484–498 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Massey, J.L.: Shift-Register Synthesis and BCH Decoding. IEEE Trans. Info. Theory 15(1), 122–127 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  • Sakata, S.: Synthesis of Two-Dimensional Linear Feedback Shift-Registers and Groebner Basis. LNCS, vol. 356. Springer, Heidelberg (1989)

    Google Scholar 

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Authors and Affiliations

  1. Fudan University, Shanghai 200433, P.R. China

    Peizhong Lu & Yan Zou

Authors
  1. Peizhong Lu
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  2. Yan Zou
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Claude Carlet Berk Sunar

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Lu, P., Zou, Y. (2007). Fast Computations of Gröbner Bases and Blind Recognitions of Convolutional Codes. In: Carlet, C., Sunar, B. (eds) Arithmetic of Finite Fields. WAIFI 2007. Lecture Notes in Computer Science, vol 4547. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73074-3_24

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  • DOI: https://doi.org/10.1007/978-3-540-73074-3_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73073-6

  • Online ISBN: 978-3-540-73074-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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