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Rank for Some Families of Quaternary Reed-Muller Codes

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Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5527))

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Abstract

Recently, new families of quaternary linear Reed-Muller codes such that, after the Gray map, the corresponding ℤ4-linear codes have the same parameters and properties as the codes in the usual binary linear Reed-Muller family have been introduced. A structural invariant, the rank, for binary codes is used to classify some of these ℤ4-linear codes. The rank is established generalizing the known results about the rank for ℤ4-linear Hadamard and ℤ4-linear extended 1-perfect codes.

This work was supported in part by the Spanish MEC and the European FEDER under Grant MTM2006-03250.

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References

  1. Bierbrauer, J.: Introduction to coding theory. Chapman & Hall/CRC, Boca Raton (2005)

    MATH  Google Scholar 

  2. Borges, J., Fernández, C., Phelps, K.T.: Quaternary Reed-Muller codes. IEEE Trans. Inform. Theory 51(7), 2686–2691 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Borges, J., Fernández-Córdoba, C., Phelps, K.T.: “ZRM codes”. IEEE Trans. Inform. Theory 54(1), 380–386 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Borges, J., Phelps, K.T., Rifà, J., Zinoviev, V.A.: On Z 4-linear Preparata-like and Kerdock-like codes. IEEE Trans. Inform. Theory 49(11), 2834–2843 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borges, J., Phelps, K.T., Rifà, J.: The rank and kernel of extended 1-perfect Z 4-linear and additive non-Z 4-linear codes. IEEE Trans. Inform. Theory 49(8), 2028–2034 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fernández-Córdoba, C., Pujol, J., Villanueva, M.: On rank and kernel of Z 4-linear codes. In: Barbero, A. (ed.) ICMCTA 2008. LNCS, vol. 5228, pp. 46–55. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  7. Fernández-Córdoba, C., Pujol, J., Villanueva, M.: ℤ2ℤ4-linear codes: rank and kernel. Discrete Applied Mathematics (2008) (submitted), arXiv:0807.4247

    Google Scholar 

  8. Hammons, A.R., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., Solé, P.: The Z 4-linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inform. Theory 40, 301–319 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  10. Krotov, D.S.: Z 4-linear Hadamard and extended perfect codes. In: International Workshop on Coding and Cryptography, Paris, France, January 8-12, pp. 329–334 (2001)

    Google Scholar 

  11. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland Publishing Company, Amsterdam (1977)

    MATH  Google Scholar 

  12. Phelps, K.T., Rifà, J., Villanueva, M.: On the additive (Z 4-linear and non-Z 4-linear) Hadamard codes: rank and kernel. IEEE Trans. Inform. Theory 52(1), 316–319 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Pernas, J., Pujol, J., Villanueva, M.: Kernel dimension for some families of quaternary Reed-Muller codes. In: Calmet, J., Geiselmann, W., Müller-Quade, J. (eds.) Beth Festschrift. LNCS, vol. 5393, pp. 128–141. Springer, Heidelberg (2008)

    Google Scholar 

  14. Pujol, J., Rifà, J., Solov’eva, F.I.: Quaternary Plotkin constructions and quaternary Reed-Muller codes. In: Boztaş, S., Lu, H.-F(F.) (eds.) AAECC 2007. LNCS, vol. 4851, pp. 148–157. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  15. Pujol, J., Rifà, J., Solovéva, F.I.: Construction of Z 4-linear Reed-Muller codes. IEEE Trans. Inform. Theory 55(1), 99–104 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Solov’eva, F.I.: On Z4-linear codes with parameters of Reed-Muller codes. Problems of Inform. Trans. 43(1), 26–32 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wan, Z.-X.: Quaternary codes. World Scientific Publishing Co. Pte. Ltd., Singapore (1997)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Dept. of Information and Communications Engineering, Universitat Autònoma de Barcelona, Spain

    Jaume Pernas, Jaume Pujol & Mercè Villanueva

Authors
  1. Jaume Pernas
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  2. Jaume Pujol
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  3. Mercè Villanueva
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Editors and Affiliations

  1. Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Avinguda Països Catalans, 26, 43007, Tarragona, Spain

    Maria Bras-Amorós

  2. Department of Mathematics, The Technical University of Denmark, Building 303, 2800, Lyngby, Denmark

    Tom Høholdt

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Pernas, J., Pujol, J., Villanueva, M. (2009). Rank for Some Families of Quaternary Reed-Muller Codes. In: Bras-Amorós, M., Høholdt, T. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2009. Lecture Notes in Computer Science, vol 5527. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02181-7_5

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  • DOI: https://doi.org/10.1007/978-3-642-02181-7_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02180-0

  • Online ISBN: 978-3-642-02181-7

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