Abstract.
The Main Conjecture on maximum distance separable (MDS) codes states that, except for some special cases, the maximum length of a q-ary linear MDS code of is q+1. This conjecture does not hold true for near maximum distance separable codes because of the existence of q-ary near-MDS elliptic codes having length bigger than q+1. An interesting related question is whether a near-MDS elliptic code may be extended to a longer near-MDS code. In this paper we prove some non-extendibility results for certain near-MDS elliptic codes.
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De Boer, M.A.: Almost MDS codes. Des. Codes Cryptogr. 9, 143–155 (1996)
S.M. Dodunekov, S.M., Landjev, I.N.: On near-MDS codes. J. Geom. 54, 30–43 (1995)
Giulietti, M.: On plane arcs contained in cubic curves. Finite Fields Appl. 8, 69–90 (2002)
Hartshorne, R.: Algebraic Geometry. Grad. Texts in Math. Vol. 52, Berlin New York: Springer Verlag, 1977
Hirschfeld, J.W.P.: Projective Geometries Over Finite Fields, 2nd edition. Oxford: Oxford University Press, 1998
Hirschfeld, J.W.P., Storme, L.: The packing problem in statistics, coding theory and finite projective planes. J. Statist. Plann. Inference 72, 355–380 (1998)
Hirschfeld J.W.P., Voloch, J.F.: The characterization of elliptic curves over finite fields. J. Austral. Math. Soc. Ser. A 45, 275–286 (1988)
Landjev, I.N.: Linear codes over finite fields and finite projective geometries. Discrete Math. 213, 211–244 (2000)
Silverman, J.H.: The Arithmetic of Elliptic Curves. Berlin Heidelberg New York Tokyo: Springer Verlag, 1986
Tsfasman, M.A., Vladut, S.G.: Algebraic-Geometric Codes. Amsterdam: Kluwer, 1991
R.J. Walker, R.J.: Algebraic Curves. Berlin Heidelberg New York: Springer Verlag, 1978
Waterhouse, W.G.: Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. 2, 521–560 (1969)
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Giulietti, M. On the Extendibility of Near-MDS Elliptic Codes. AAECC 15, 1–11 (2004). https://doi.org/10.1007/s00200-003-0141-5
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DOI: https://doi.org/10.1007/s00200-003-0141-5