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On the Feng-Rao numbers

Some computations concerning the so-called Feng-Rao numbers are presented. Such numbers, for some numerical semigroup S and positive integer r fixed, describe the different asymptotical behaviour of the generalized Feng-Rao distances in S with respect to the classical Feng-Rao distance, whose asymptotical behaviour is essentially related to the Goppa distance. These concepts naturally arise from the theory of Algebraic Geometry codes for the case of Weierstrass semigroups, but here they are studied in general for any given numerical semigroup. The main results of this paper are addressed to the computation of the Feng-Rao numbers for simple semigroups, such as those with multiplicity two and those generated by {a, a + 1} with a a positive integer.