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Gröbner Bases for the Distance Distribution of Systematic Codes

In: Gröbner Bases, Coding, and Cryptography

Author

Listed:
  • Eleonora Guerrini

    (University of Trento, Department of Mathematics)

  • Emmanuela Orsini

    (University of Pisa, Department of Mathematics)

  • Ilaria Simonetti

    (University of Milan, Department of Mathematics)

Abstract

Coding theorists have been studying only linear codes, with a few exceptions (Preparata in Inform. Control 13(13):378–400, 1968; Baker et al. in IEEE Trans. on Inf. Th. 29(3):342–345, 1983). This is not surprising, since linear codes have a nice structure, easy to study and leading to efficient implementations. However, it is well-known that some non-linear codes have a higher distance (or a better distance distribution) that any linear code with the same parameters (Preparata in Inform. Control 13(13):378–400, 1968; Pless et al. (eds.) in Handbook of Coding Theory, vols. I, II, North-Holland, Amsterdam, 1998). This translates into a superior decoding performance (Litsyn in Handbook of Coding Theory, vols. I, II, North-Holland, Amsterdam, pp. 463–498, 1998). Systematic non-linear codes are the most studied non-linear codes. We describe a Gröbner bases technique to compute the distance distribution for these codes.

Suggested Citation

  • Eleonora Guerrini & Emmanuela Orsini & Ilaria Simonetti, 2009. "Gröbner Bases for the Distance Distribution of Systematic Codes," Springer Books, in: Massimiliano Sala & Shojiro Sakata & Teo Mora & Carlo Traverso & Ludovic Perret (ed.), Gröbner Bases, Coding, and Cryptography, pages 367-372, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-93806-4_22
    DOI: 10.1007/978-3-540-93806-4_22
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