On the Goppa morphism

Abstract: We investigate the geometric foundations of the space of geometric Goppa codes using the Tsfasman-Vladut H-construction. These codes are constructed from level structures, which extend the classical Goppa framework by incorporating invertible sheaves and their trivializations over rational points. A key contribution is the definition of the Goppa morphism, a map from the universal moduli space of level structures, denoted $LS_{g,n,d}$, to certain Grassmannian $\mathrm{Gr}(k,n)$. This morphism allows problems related to distinguishing attacks and key recovery in the context of Goppa Code-based Cryptography to be translated into a geometric language, addressing questions about the equations defining the image of the Goppa morphism and its fibers. Furthermore, we identify the ranges of the degree parameter $d$ that should be avoided to maintain security against distinguishers. Our results, valid over arbitrary base fields, also apply to convolutional Goppa codes.