Explicit Computation of Generalized Hamming Weights for Some Algebraic Geometric Codes

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The generalized Hamming weights, introduced a few years ago by V. K. Wei, provide substantial information of codes and thus play a central role in coding theory. For algebraic geometric codes, there have been many works on their generalized Hamming weights (or weight hierarchy). However, for lots of codes from Hermitian curves and the Klein quartic, some generalized Hamming weights still have not yet been found explicitly. In this paper, we first prove a general result (Theorem 1.4) on the computation of generalized Hamming weights of geometric Goppa codes on plane curves, using the configuration ofFq-rational points on the curves. Then we give the exact values (Theorem 2.2) of the first and second generalized Hamming weights of some codes arising from the Klein quartic. Our main result (Theorem 2.3) gives the exact values of the second and third generalized Hamming weights of certain codes from Hermitian curves. In the Appendix, a previous known result of Yang, Kumar, and Stichtenoth for Hermitian codes is shown to follow from Theorem 1.4. We also give the exact values of the first three generalized Hamming weights for Fermat codes.