Résumé
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.