Generalized Hamming Weights of Duals of Algebraic-geometric Codes

Here we extends a work of A. Couvreur on the Hamming distance of the dual of an evaluation code to its generalized Hamming weights. We prove the following result. Fix integers r ≥ 2, m > 0 and e ≥ 1. Let Z ⊂ P be a zero-dimensional scheme such that deg(Z) ≤ 3m + r − 3. If r > 2 assume that Z spans P and that the sum of the degrees of the non-reduced connected components of Z is at most 2m+ 1. We have h(IZ(m)) ≥ e if and only if there is W ⊆ Z as one of the schemes in the following list: (a) deg(W ) = m+ 1 + e and W is contained in a line; (b) deg(W ) = 2m+ 1 + e and W is contained in a reduced plane conic; (c) r ≥ 3, e ≥ 2, and there are an integer f ∈ {1, . . . , e− 1} and lines L1, L2, such that L1 ∩ L2 = ∅, deg(L1 ∩ Z) = m + 1 + f and deg(L2 ∩ Z) = m+ 1 + e− f . AMS Subject Classification: 4N05, 14Q05, 94B27