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.
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References
R. D. Baker, J. H. van Lint and R. M. Wilson, On the Preparata and Goethals codes, IEEE Trans. on Inf. Th. 29 (1983), no. 3, 342–345.
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Guerrini, E., Orsini, E., Simonetti, I. (2009). Gröbner Bases for the Distance Distribution of Systematic Codes. In: Sala, M., Sakata, S., Mora, T., Traverso, C., Perret, L. (eds) Gröbner Bases, Coding, and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93806-4_22
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DOI: https://doi.org/10.1007/978-3-540-93806-4_22
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