The second Feng-Rao number of every inductive numerical semigroup is explicitly computed. This number determines the asymptotical behavior of the order bound for the second Hamming weight of one-point algebraic geometry codes. In particular, this result is applied for the codes defined by asymptotically good towers of function fields whose Weierstrass semigroups are inductive. In addition, some properties of inductive numerical semigroups are studied, the involved Apéry sets are computed in a recursive way, and some tests to check whether given numerical semigroups are inductive or not are provided.

The weight hierarchy of one-point algebraic geometry codes can be estimated by means of the generalized order bounds, which are described in terms of a certain Weierstrass semigroup. The asymptotical behavior of such bounds for$r\geq 2$differs from that of the classical Feng–Rao distance$(r=1)$by the so-called Feng–Rao numbers. This paper is addressed to compute the Feng–Rao numbers for numerical semigroups of embedding dimension two (with two generators), obtaining a closed simple formula for the general case by using numerical semigroup techniques. These involve the computation of the Apéry set with respect to an integer of the semigroups under consideration. The formula obtained is applied to lower bounding the generalized Hamming weights, improving the bound given by Kirfel and Pellikaan in terms of the classical Feng–Rao distance. We also compare our bound with a modification of the Griesmer bound, improving this one in many cases.

We compute the order (or Feng-Rao (1994)) bound on the minimum distance of one-point algebraic-geometry codes C/sub /spl Omega//(P, /spl rho//sub t/Q), when the Weierstrass semigroup at the point Q is an Arf 91949) semigroup. The results developed to that purpose also provide the dimension of the improved geometric Goppa codes related to these C/sub /spl Omega// (P, /spl rho//sub t/Q).