On the duals of geometric Goppa codes from norm-trace curves |
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Institution: | 1. Dept. of Mathematics, University of Trento, Via Sommarive 14, 38123 Povo (TN), Italy;2. Dept. of Mathematics, University of Neuchâtel, Rue Emile-Argand 11, CH-2000 Neuchâtel, Switzerland |
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Abstract: | In this paper we study the dual codes of a wide family of evaluation codes on norm-trace curves. We explicitly find out their minimum distance and give a lower bound for the number of their minimum-weight codewords. A general geometric approach is performed and applied to study in particular the dual codes of one-point and two-point codes arising from norm-trace curves through Goppaʼs construction, providing in many cases their minimum distance and some bounds on the number of their minimum-weight codewords. The results are obtained by showing that the supports of the minimum-weight codewords of the studied codes obey some precise geometric laws as zero-dimensional subschemes of the projective plane. Finally, the dimension of some classical two-point Goppa codes on norm-trace curves is explicitely computed. |
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