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A vector version of the BMS algorithm for implementing fast erasure-and-error decoding of one-point AG codes

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Book cover Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1255))

Abstract

In this paper1 we propose a fast erasure-and-error decoding method for any one-point Algebraic-Geometric (AG) codes up to the Feng-Rao bound, which is a modification of the Sakata algorithm (the nD Berlekamp-Massey algorithm) and is given as a unification of our previous fast error-only decoding method and the algorithm for finding a minimal polynomial vector set of a given vector of nD arrays.

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References

  1. S. Sakata, “Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array”, J. Symbolic Computation, vol. 5, pp. 321–337, 1988.

    Google Scholar 

  2. S. Sakata, “Extension of the Berlekamp-Massey algorithm to N dimensions”, Information and Computation, vol. 84, pp. 207–239, 1990.

    Google Scholar 

  3. S. Sakata, J. Justesen, Y. Madelung, H. Elbrønd Jensen, and T. Høholdt, “A fast decoding method of AG codes from Miura-Kamiya curves C ab up to half the Feng-Rao bound”, Finite Fields and Their Applications, vol. 1, pp. 83–101, 1995.

    Google Scholar 

  4. S. Sakata, J. Justesen, Y. Madelung, H. Elbrønd Jensen, and T. Høholdt, “Fast decoding of algebraic geometric codes up to the designed minimum distance”, IEEE Transactions on Information Theory, vol. 41, Part I, pp. 1672–1677, 1995.

    Google Scholar 

  5. S. Sakata, “nD shift register synthesis on convex cones and cylinders and fast decoding of general one-point AG codes”, presented at the 1994 IEEE Workshop on Information Theory, Moscow, Russia, July 1994; Bulletin of the University of Electro-Communications, vol. 8, pp. 187–203, 1995.

    Google Scholar 

  6. S. Sakata, H. Elbrønd Jensen, and T. Høholdt, “Generalized Berlekamp-Massey decoding of algebraic geometric codes up to half the Feng-Rao bound”, IEEE Transactions on Information Theory, vol. 41, Part I, pp. 1762–1768, 1995.

    Google Scholar 

  7. G.L. Feng and T.R.N. Rao, “Decoding algebraic-geometric codes up to the designed minimum distance”, IEEE Transactions on Information Theory, vol. 39, pp. 37–45, 1993.

    Google Scholar 

  8. T. Høholdt and R. Pellikaan, On the decoding of Algebraic Geometry codes, IEEE Transactions on Information Theory, vol. 41, Part I, pp. 1589–1614, 1995.

    Google Scholar 

  9. G.D. Forney, “On decoding BCH codes”, IEEE Transactions on Information Theory, vol. 11, pp. 549–557, 1965.

    Google Scholar 

  10. G.L.Feng and T.R.N. Rao, “Erasure-and-error decoding of algebraic-geometric codes”, presented at the 1993 IEEE Information Theory Workshop, Shizuoka, Japan, June 1993.

    Google Scholar 

  11. S. Sakata, “Finding a minimal polynomial vector of a vector of nD arrays”, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Proceedings of AAECC-9, New Orleans, USA: Lecture Notes in Computer Science, 539, Springer Verlag, pp.414–425, 1991.

    Google Scholar 

  12. R. Kötter, On Algebraic Decoding of Algebraic Geometric and Cyclic Codes, Dessertation, Department of Electrical Engineering, Linköping, Sweden, 1996.

    Google Scholar 

  13. D.A. Leonard, “Error-locator ideals for algebraic-geometric codes”, IEEE Transactions on Information Theory, vol. 41, Part I, pp. 819–824, 1995.

    Google Scholar 

  14. A.N. Skorobogatov and S.G. Vlăduţ, “On the decoding of algebraic-geometric codes”, IEEE Transactions on Information Theory, vol. 36, pp. 1051–1060, 1990.

    Google Scholar 

  15. S. Sakata and M. Kurihara, “A systolic array architecture for implementing a fast parallel decoding algorithm of one-point AG-codes”, submitted for presentation at the 1997 IEEE International Symposium on Information Theory, Ulm, Germany, July 1997.

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Information Mathematics, The University of Electro-Communications, Chofu-shi, 182, Tokyo, Japan

    Shojiro Sakata

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  1. Shojiro Sakata
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Teo Mora Harold Mattson

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© 1997 Springer-Verlag Berlin Heidelberg

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Sakata, S. (1997). A vector version of the BMS algorithm for implementing fast erasure-and-error decoding of one-point AG codes. In: Mora, T., Mattson, H. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1997. Lecture Notes in Computer Science, vol 1255. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63163-1_23

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  • DOI: https://doi.org/10.1007/3-540-63163-1_23

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63163-7

  • Online ISBN: 978-3-540-69193-8

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