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

## TL;DR

This paper analyzes the second Feng-Rao distance in algebraic geometry codes linked to Arf semigroups, providing bounds and computations that enhance understanding of code parameters and their relation to semigroup properties.

## Contribution

It introduces a method to compute the second Feng-Rao distance for Arf semigroups and applies it to codes from asymptotically good towers, expanding the theoretical framework.

## Key findings

- Provides a lower bound for the second Hamming weight of AG codes.
- Calculates the second Feng-Rao number for specific semigroups.
- Includes examples and comparisons with previous results.

## 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 Ap\'{e}ry sets, and thus several results concerning Ap\'ery sets of Arf semigroups are presented.