On the weight hierarchy of geometric Goppa codes

Abstract

The weight hierarchy of a linear code is the set of generalized Hamming weights of the code. In the paper, the authors consider geometric Goppa codes and provide a lower bound on their generalized Hamming weights similar to Goppa's lower bound on their minimum distance. In the particular case of Hermitian codes, exact results on the second and third generalized Hamming weights are given for any m except a few cases, where m is a parameter that governs the dimension of these codes. In many instances, the authors are able to provide considerably more information on their generalized Hamming weights. An upper bound relating the generalized Hamming weights of Hermitian codes to the pole numbers at a special point on the curve is also provided. Similar results are given in the case of codes from some subfields of the Hermitian function fields, which are also maximal. Finally, a nontrivial family of codes is also presented whose weight hierarchy is completely determined.