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Codes over Z 4

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Computational Discrete Mathematics

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2122))

Abstract

One recent direction in coding theory has been to study linear codes over the alphabet Z 4 and apply the Gray map from Z 4 to binary pairs to obtain binary nonlinear codes better than comparable binary linear codes. This connection between linear codes over Z 4 and nonlinear binary codes was also the breakthrough in solving an old puzzle of the apparent duality between the nonlinear Kerdock and Preparata codes. We present a description of this puzzle and a brief introduction to codes over Z 4.

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References

  1. A.R. Calderbank and G. McGuire, “Construction of a (64, 237,12) code via Galois rings”, Designs, Codes and Cryptography, 10 (1997), 157–165.

    Article  MATH  MathSciNet  Google Scholar 

  2. A.R. Calderbank, G. McGuire, P.V. Kumar and T. Helleseth, “Cyclic codes over Z 4, locator polynomials and Newton’s identities”, IEEE Trans. Inform. Theory, 42 (1996), 217–226.

    Article  MATH  MathSciNet  Google Scholar 

  3. A.R. Hammons Jr., P.V. Kumar, A.R. Calderbank, N.J.A. Sloane and P. Solé, “The Z4-linearity of Kerdock, Preparata, Goethals, and related codes”, IEEE Trans. Inform. Theory, 40 (1994), 301–319.

    Article  MATH  MathSciNet  Google Scholar 

  4. T. Helleseth, P. V. Kumar and A. Shanbhag, “Exponential sums over Galois rings and their applications”, Finite Fields and their Applications, London Mathematical Society, Lecture Note Series 233, edited by S. Cohen and H. Niederreiter,Cambridge University Press, (1996), 109–128.

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  5. A.M. Kerdock, “A class of low-rate nonlinear binary codes”, Inform. Contr., 20 (1972), 182–187.

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  6. A. Nechaev, “The Kerdock code in a cyclic form”, Discrete Math. Appl., 1 (1991), 365–384.

    Article  MATH  MathSciNet  Google Scholar 

  7. F.P. Preparata, “A class of optimum nonlinear double-error correcting codes”, Inform. Contr., 13 (1968), 378–400.

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Author information

Authors and Affiliations

  1. Department of Informatics, University of Bergen, Høyteknologisenteret, N-5020, Bergen, Norway

    Tor Helleseth

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  1. Tor Helleseth
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Editor information

Editors and Affiliations

  1. Institut für Informatik, Freie Universität Berlin, Takustr. 9, 14195, Berlin, Germany

    Helmut Alt

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© 2001 Springer-Verlag Berlin Heidelberg

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Helleseth, T. (2001). Codes over Z 4 . In: Alt, H. (eds) Computational Discrete Mathematics. Lecture Notes in Computer Science, vol 2122. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45506-X_4

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  • DOI: https://doi.org/10.1007/3-540-45506-X_4

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42775-9

  • Online ISBN: 978-3-540-45506-6

  • eBook Packages: Springer Book Archive

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