Goppa-like bounds for the generalized Feng-Rao distances

We give some general bounds and formulas for the generalized Feng-Rao distances (or generalized order bounds) in an arbitrary numerical semigroup. The obtained results can be regarded as generalizations of well-known facts on the classical Feng-Rao distance (or first order bound), namely its connection with the Goppa distance. These results show that their asymptotical behaviour is essentially the same as in the case of the classical order bound. Explicit computations are given for the second Feng-Rao distance.