A study of the $m$th relative Feng-Rao distances with a special emphasis on $C_{ab}$ curves

In the seminal paper [13], Farrán and Munuera introduced a method to establish generic information on the $m$th Feng-Rao distance via a function on arbitrary numerical semigroups that serves as a lower bound on the generalized Hamming weights of dual one-point algebraic geometry codes whenever the semigroup is Weierstrass. In the present paper, we study what we shall call the primary and dual, respectively, $m$th relative Feng-Rao distances, which serve as lower bounds on relative generalized Hamming weights of nested primary and dual, respectively, one-point codes under similar condition. A unified method for establishing generic formulas for $m$th relative Feng-Rao distances of general numerical semigroups is developed and for a large variety of numerical semigroups generated by two elements, closed formula expressions are established complementing Delgado et al.s work [8] on the $m$th Feng-Rao distances of such semigroups. Finally, for non-singular $C_{ab}$ curves, we develop a method to estimate the $m$th relative generalized Hamming weight of dual and primary one-point codes in those regions where the generic methods may fail to provide best possible results.