Goppa-like AG codes from $C_{a,b}$ curves and their behaviour under squaring their dual

In this paper, we introduce a family of codes that can be used in a McEliece cryptosystem, called Goppa--like AG codes. These codes generalize classical Goppa codes and can be constructed from any curve of genus $\mathfrak{g} \geq 0$. Focusing on codes from $C_{a,b}$ curves, we study the behaviour of the dimension of the square of their dual to determine their resistance to distinguisher attacks similar to the one for alternant and Goppa codes developed by Mora and Tillich. We also propose numerical experiments to measure how sharp is our bound.