On the second Feng-Rao distance of Algebraic Geometry codes related to Arf semigroups

Abstract

We describe the second (generalized) Feng-Rao distance for elements in an Arf numerical semigroup that are greater than or equal to the conductor of the semigroup. This provides a lower bound for the second Hamming weight for one point AG codes. In particular, we can obtain the second Feng-Rao distance for the codes defined by asymptotically good towers of function fields whose Weierstrass semigroups are inductive. In addition, we compute the second Feng-Rao number, and provide some examples and comparisons with previous results on this topic. These calculations rely on Apéry sets, and thus several results concerning Apéry sets of Arf semigroups are presented.

Ideals of Numerical Semigroups and Error-Correcting Codes

“…Farrán and Munuera showed the existence of a constant, which they named the Feng-Rao number, depending only on the dimension of the Hamming weights and the Weierstrass semigroup, which completely determined the order bounds for codes of rate low enough. The references [41][42][43][44] deal with the generalized order bounds and the Feng-Rao numbers related to particular classes of semigroups.…”