On numerical semigroups and the redundancy of improved codes correcting generic errors

We introduce a new sequence τ associated to a numerical semigroup similar to the ν sequence used to define the order bound on the minimum distance and to describe the Feng–Rao improved codes. The new sequence allows a nice description of the optimal one-point codes correcting generic errors and to compare them with standard codes and with the Feng–Rao improved codes. The relation between the τ sequence and the ν sequence gives a new characterization of Arf semigroups and it is shown that the τ sequence of a numerical semigroup unequivocally determines it.